Harmonic Analysis on Classical Groups

Proceedings of the Analysis Conference, Singapore 1986 S.T.L. Choy, J.P. Jesudason, P.Y. Lee (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1988

69

HARMONIC ANALYSIS ON CLASSICAL GROUPS Gony Sheny The C h i n e s e hiv e r s i t y o f S c i e n c e and Techno1 o y y L i S h i X i o n g and Zheny Xue An D e p a r t m e n t o f M a t h e m a t i c s , Anhui U n i v e r s i t y , H e f e i , P e o p l e ' s R e p u b l i c o f China The p u r p o s e o f t h i s a r t i c l e i s t o i n t r o d u c e b r i e f l y t h e p r i n c i p a l r e s u l t s i n h a r m o n i c a n a l y s i s on c l a s s i c a l g r o u p s and i t s e x t e n s i o n on compact L i e g r o u p s i n China, and a l s o t o i n t r o d u c e b r i e f l y some i m p o r t a n t r e s u l t s i n t h i s d i r e c t i o n abroad.

P r o f . Hua Luo Geny h a v i ny accompl ished h i s famous work "Harmonic A n a l y s i s on C l a s s i c a l Domains i n t h e Theory o f F u n c t i o n s o f S e v e r a l Complex V a r i a b l e s " , a p p l i e d h i s t h e o r y t o t h e h a r m o n i c a n a l y s i s on u n i t a r y yroups,

deepened t h e

w e l l - k n o w n Peter-Weyl T h e o r m and i n i t i a t e d t h e r e s e a r c h on h a r m o n i c a n a l y s i s on c l a s s i c a l y r o u p s i n C h i n a [l]. I n t h e l a t e 1 9 5 0 ' s , based on H u a ' s work a s y s t e m a t i c r e s e a r c h on h a r m o n i c a n a l y s i s on u n i t a r y y r o u p s was c a r r i e d on w e l l [2]-[6].

A s e r i e s o f concepts,

d e f i n i t i o n s and methods were s e t up, w h i c h s u b s e q u e n t l y i n f l u e n c e d t h e r e s e a r c h f o r h a r m o n i c a n a l y s i s on c l a s s i c a l g r o u p s and compact L i e yroups. t h a t t h e r e s e a r c h was suspended f o r a l o n y t i m e .

It i s a p i t y

The r e a s o n i s now w e l l - k n o w n .

I n t h e l a t e 1970's, t h e research regained i t s s t r e n g t h i n China.

The b a s i c

i d e a i s t h a t u n i t a r y y r o u p s a r e t h e c h a r a c t e r i s t i c m a n i f o l d s f o r t h e complex c l a s s i c a l domains o f t h e f i r s t c l a s s and b o t h r o t a t i o n g r o u p s and u n i t a r y s y m p l e c t i c y r o u p s a r e c h a r a c t e r i s t i c m a n i f o l d s f o r r e a l c l a s s i c a l domains o f t h e f i r s t c l a s s and f o r t h e c l a s s i c a l domains o f t h e q u a t e r n i o n s r e s p e c t i v e l y . Chen Guany X i a o From t h i s p o i n t o f view, Wany Shi Kun, Dong Dao Zhen, He Zu Qi, and o t h e r s s y s t e m a t i c a l l y s t u d i e d t h e h a r m o n i c a n a l y s i s on r o t a t i o n y r o u p s and u n i t a r y s y m p l e c t i c yroups.

L a t e r L i S h i Xiony,

Zheny Xue An, Fan Da Shan, Chen

Shun Fu c o n t i n u e d t h i s r e s e a r c h and e x t e n d e d i t t o compact L i e g r o u p s .

A t p r e s e n t t h e r e s e a r c h i n t h i s d i r e c t i o n i s c a r r i e d on w e l l . S i n c e t h e l a t e 1 9 6 0 ' s , on t h e o t h e r hand, many r e s e a r c h e r s a b r o a d have s t u d i e d t h e h a r m o n i c a n a l y s i s on compact L i e y r o u p s such as E. M. S t e i n , R. Coifman and G. Weiss, J. L. C l e r c , R. J . S t a n t o n and P. A. Tomas, R.

S.

N. J . Weiss, D. L. R a y o z i n , 6. D r e s e l e r , R. A. Mayer, M. E. T a y l o r , M. S u y i u r a , S. G i u l i n , P. M. Soard, G. T r a v a y l i n , 6. George, H. Johuen and Strichartz, others.

S. Gong et al.

I0

The p u r p o s e o f t h i s a r t i c l e i s t o i n t r o d u c e b r i e f l y t h e p r i n c i p a l r e s u l t s i n harmonic a n a l y s i s on c l a s s i c a l y r o u p s and i t s e x t e n s i o n on compact L i e g r o u p s i n China, and a l s o t o i n t r o d u c e b r i e f l y some i m p o r t a n t r e s u l t s i n t h i s d i r e c t i o n abroad.

The r e l a t e d p r o o f s a r e o m i t t e d .

1. Poisson Kernels and Abel Sumnation The r e s e a r c h i n h a r m o n i c a n a l y s i s on u n i t a r y y r o u p s was i n i t i a t e d f r o m t h e P o i s s o n k e r n e l s on t h e c l a s s i c a l domains o f s e v e r a l complex v a r i a b l e s d e f i n e d by Hua Luo Geny. L e t RI b e t h e c l a s s i c a l domain c o n s i s t i n y o f a l l complex m a t r i c e s o f o r d e r n such t h a t

- Zll

I

> u.

It i s we1 -known t h a t t h e c h a r a c t e r i s t i c m a n i f o l d s o f RI

are t h e u n i t a r y yroups

Un o f o r d r n, and t h e e l e m e n t i n t h e a n a l y t i c automorphism g r o u p can be r e p r e -

s e n t e d by

W = (AZ + B)(CZ + U ) - l where W , Z E RI and 2nx2n m a t r i x

F = ( : satisfies the followiny three conditions :

or)

d e t F = 1. On Un,

(1.1) chanyes i n t o V = (AU + B)(CU + O)-'

(1.4)

w h i c h t r a n s f o r m s a u n i t a r y m a t r i x U i n t o a n o t h e r u n i t a r y m a t r i x V. Let

fi

and

\i

d e n o t e t h e r e s p e c t i v e volume e l e m e n t s o f U and V,

\j = ( d e t ( C U + D ) L e t a p o i n t Z o f RI become 0 u n d e r (1.1) become V.

then

I-2n fi. and, a c c o r d i n g l y ,

(1.5) a p o i n t U o f Un

Then Hua Luo Geng s t a r t i n y from t h e t h e o r y o f harmonic f u n c t i o n s i n

s e v e r a l complex v a r i a b l e s , d e f i n e d t h e P o i s s o n K e r n e l as f o l l o w s : P(Z,U)

=

det(1 ldet(Z

-

-

ZT')n U)lZn

Harmonic Analysis on Classical Groups

71

and p r o v e d t h e f o l l o w i n y L e t $ ( U ) be a c o n t i n u o u s f u n c t i o n on Un,

THEOREM 1.1.

The P o i s s o n k e r n e l P(r1,U)

on u n i t a r y g r o u p s i n (1.7)

then

has t h e f o l l o w i n y

expansion P(r1.U)

fz,

where N ( f ) = (fl,

..., f n )

=

1 pf(r)N(f)xf(U), f

i s the order of the sinyle-valued irreducible

u n i t a r y r e p r e s e n t a t i o n A f ( U ) o f Un w h i c h t a k e s f = ( f l , labels ( f l > f2 > characters,

... > f,

..., f n )

f2,

as i t s

a l l are i n t e y e r s ) , xf(U) are t h e correspondiny

and P f ( d

in

P(rI,lJ)xf(U)fi.

=

(1.8)

I f u(U) i s an i n t e g r a b l e f u n c t i o n on Un and i t s F o u r i e r s e r i e s i s u(U) where

-F

c f = w

N(f)tr(CfAf(U)),

I

u(U)Af(U')fi.

'n Then t h e Abel sum o f (1.9)

is (1.10)

pf(r)N(f)tr(CfAf(U)). The c o n c r e t e f o r m u l a f o r p f ( r ) i s i n c l u d e d i n t h e f o l l o w i n g theorem.

THEOREM 1.2.

el > e2 >

... >

pf(r) = r X

where Ns(a,b)

I f e l = fl+n-l,

[Z]

es

0 > E,+~

>

..., e k = f k + n - k , ..., en = ... > en ( n > s > 0 ) , we have

fn, when

fl+...+fs-fs+l-...-fn

N( f ,g)N( y , f ) N ( f ) N ( Y)

I:

s>gs+l>...'y,>o

= N(a1,

...,as,bs+l, ...,b n ) ,

g = (yl

,...,yn)

(1.11)

can a l s o be w r i t t e n as

a r e a p e r m u t a t i o n o f 0,1,2

s

X

n-l>vs+l>...>v

I:

r2(gs+1+"'+gn)

Y1+n-l and gZ+n-Z

,...,n - 1

,...,gn

and O>y1>gz>

(e.-vk)(v.-ek) J j = 1 k=l (v.-v ) ( e . - e ) >O J k J k n

n

(1.11)

I

n

n

in

...>ys,s-n.

Z(V~+~+...+V~) r

S.Gong et al.

12

The p r o o f o f Theorem 1.2 i s c o m p l i c a t e d and needs h i g h l y s k i l l f u l c a l c u l a tion.

Here a s k e t c h i s y i v e n and r e a d e r s a r e r e f e r r e d t o [ 2 ]

for details.

We i n t r o d u c e t h e f o l l o w i n g n o t a t i o n s

where q,t

a r e n o n - n e g a t i v e i n t e g e r s , and p i s an i n t e y e r .

I f q = U and p < 0 , t h e n we have

( s i n c e eipe(l-re-ie)-t powers.

"'st

)

= 0,

P < 0,

for

(1.13)

i s a F o u r i e r s e r i e s whose t e r m s have o n l y n e g a t i v e

Similarly,

( qu;

)

for

= 0,

p

> U.

(1.14)

I n virtue of e ipe

( l-rei e)q( l-re-i

lt

( l-re-i')t

( l-rei

,,i ( p + 1 )e

-

,ipe

( l-reie ) q ( l - r e - i ' ) t

'

we have

I n t h e same way, we have

(

q,t Again, f r o m

,e-,

I*

(1.16)

i t i s deduced t h a t

(1.15),

(1.16)

and (1.17) a r e t h r e e b a s i c r u l e s o f c a l c u l a t i o n .

a p p l i c a t i o n o f (1.15)

( q P, t

Repeated

leads t o =

=

-

( q - Pl , t

1

r(

P

( q-2,t

.....

$1 P+1 )

P+l

+

( q,t

r( q-1,t

I f p < 0 , t h e n i t i s easy t o see from (1.13)

that (1.18)

Harmonic Analysis on Classical Groups

13

r e p e a t e d l y , we can o b t a i n

By u s i n y (1.18)

I f p < 0, t h e n t h e r e e x i s t s

( Similarly,

q;t

(

of

(

1.

oyq ) P.4

r-p

q-'

1

k=U

(

i t can be deduced f r o m (1.16)

I n v i r t u e o f (1.1Y) of

1=

and (1.2U),

(

P$ 0

).

)(

0

q-k,t

1'

(1.1Y)

t h a t i f p > 0, we h a v e

t h e c a l c u l a t i o n o f (1.12)

F u r t h e r m o r e , by ( 1 . 1 7 ) , and

k-p-1 k

i s reduced t o t h a t

t h e c a l c u l a t i o n can be reduced t o t h a t

On t h e o t h e r hand, i t i s easy t o see t h a t

(

U p,u

=

(

u,q

) = 1 .

(1.21)

The f o r m u l a e m e n t i o n e d above b e i n y a p p l i e d t o (1.8),

a c o m p l i c a t e d and h i y h l y

s k i l l f u l c a l c u l a t i o n can y i e l d s t h a t

x e

i(k

e +...+ knen -iB1 1 1 D(e

..., e

,

-ien

)dol

... den

(1.22)

1 ................ n ,n By u s i n y ( 1 . 1 5 ) ,

(1.16)

and (1.17)

d e t e r m i n a n t can be c a l c u l a t e d o u t . I n v i r t u e o f Theorem 1.2,

1

n,n

1

repeatedly, t h e value of t h e preceding

Thus t h e c o n c l u s i o n o f t h e t h e o r e m f o l l o w s .

t h e F o u r i e r s e r i e s (1.10)

a b s o l u t e l y converyent, t h e r e f o r e

kn-( n-1)

..-(

of

'j

u(V)P(rU,V)

\j i s

"n (1.23)

S. Gong et al.

74

From Theorein 1.1, i t f o l l o w s t h a t U(U)

=

l i m J u(V)P(rU,V) r + l lln

= liin

3

1 pf(r)N(f)tr(CfAf(U)).

r+l f

Thus t h e F o u r i e r s e r i e s o f u(U) i s Abel-summable t o i t s e l f . E v i d e n t l y , as f a r as a p p l i c a t i o n i s concerned a c o n c r e t e t h e o r e m on c o n v e r yence i s s u p e r i o r t o an a b s t r a c t e x i s t e n c e theorem on a p p r o x i m a t i o n .

Thus

Theorem 1.1 sharpens t h e famous Peter-Weyl Theorem. Assuminy t h a t u ( U ) has s u f f i c i e n t smoothness, we can deduce t h e d i f f e r e n c e between SN =

1

N > f > f >...>f 1 2

n

>-N

pf( r)N(f)tr(CfAf

(u))

I n a d d i t i o n , t h i s p r o v e s , i n t h e meantime, t h a t t h e f u n c t i o n system

and u ( U ) .

{ a i J ( U ) } c o n s i s t i n y o f a l l elements o f t h e m a t r i c e s o f t h e s i n g l e - v a l u e d i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n s A f ( U ) = ( a f. ( U ) ) f o r a u n i t a r y y r o u p i s complete.

1J

As a c o r o l l a r y , we can i m m e d i a t e l y deduce t h e a p p r o x i m a t i o n t h e o r e m

f o r any compact y r o u p and any compact homoyeneous space. L e t us c o n s i d e r t h e r e a l c l a s s i c a l domain Rn c o n s i s t i n y o f a l l r e a l m a t r i c e s o f o r d e r n such t h a t

I

-

XX' > 0,

t h e c h a r a c t e r i s t i c m a n i f o l d o f w h i c h i s o r t h o y o n a l y r o u p O(n) =

{r, rr'

= I}.

The a n a l y t i c autoinorphism

w o f Rn maps o n t o i t s e l f R,

= (AX

+ B)(cx+D)-~

O ( n ) and SO(n) o n t o Rn,

(1.24) O ( n ) and SO(n) r e s p e c t i v e l y .

Lu U i Keny w i t h t h e h e l p o f t h e t h e o r y o f h a r m o n i c f u n c t i o n s i n s e v e r a l r e a l v a r i a b l e s , d e f i n e d P o i s s o n k e r n e l s on Rn as f o l l o w s : (1.25)

As i n Theorem 1.1, we can p r o v e t h e f o l l o w i n y .

THEOREM 1.3.

[l] I f

u(r)

i s a c o n t i n u o u s f u n c t i o n on r o t a t i o n y r o u p SO(n),

then

I n t h e e a r l y 1 9 6 0 ' s , Zhony J i a (ling, u s i n g t h e method o f y e n e r a t i n y f u n c t i o n , p r o v e d t h e e x p a n s i o n o f P o i s s o n k e r n e l s on r o t a t i o n groups.

THEOREM 1.4.

[7] The P o i s s o n k e r n e l P ( r r , r ) o f r o t a t i o n g r o u p s has t h e

Harmonic Analvsis on Classical Groups

75

(1.26)

(1.27) or, e q u i v a l e n t t o

(n-2) ( r ) I

I n (1.27),

..., q,,-k-l

when n = 2 k + l , we t a k e (q1,q2,

1)

n - k - 1 > 41 >

2)

qi

... > q n - k - l

...

(n-2) ( r ) 5,-1

) from

6

which s a t i s f i e s

> 0,

+ q j # i+j-1, f o r a l l i # j ,

and

n-k-1

1

if

-1,

1

if

n-k-1

qi =

u,

qi,=

3, 4

(mod 4 ) ;

when n = 2k, i t i s t a k e n f r o m E w h i c h s a t i s f i e s

...

1)

n-k > q1 >

2)

qi

# i f o r a l l i,

3)

qi

+ qJ. # i+j f o r any i

and

E(qlS

Moreover, N(m1,

...,mk,

qn-k-1

. * . 9

91,

"3

#

j

qn-k-l ) = (-l)(ql+'"+qn-k-l)/'

...,q n - k - l )

i s t h e o r d e r of t h e i r r e d u c i b l e u n i t a r y

r e p r e s e n t a t i o n o f a u n i t a r y group o r o r d e r n-1 which takes (ml, qn-k-l)

ml

SentatiOn

ml

>

> m2 >

as i t s l a b e l s , inl

> m2 >

... > rnk = 0,

O f

... > mk

> 0.

...,mk,ql ,...,

I f n = 2 k + l o r n = 2k and

t h e n um(r) i s t h e c h a r a c t e r o f t h e i r r e d u c i b l e r e p r e -

so(n) which takes

... > m k >

(1.2Y)

(r) m

0, t h e n u

Ill

= (ml,...,mk)

as i t s l a b e l s .

1f

= 2k and

i s t h e sum o f t w o c h a r a c t e r s o f t h e i r r e d u c i b l e

r e p r e s e n t a t i o n s o f s o ( n ) w h i c h t a k e s (Inl,

...,*Ink)

as i t s l a b e l s .

S. Gong el al.

16

............ Sn-l(r) =

+

,2n-2k-3

for

n = 2k+l,

-

,2n-2k-4

for

n = 2k.

(1.30)

The p r o o f o f Theoren 1.4 i s c o m p l i c a t e d and needs h i g h l y s k i l l f u l c a l c u l a t i o n .

F o r d e t a i l s , see [7]. I-et us c o n s i d e r a domain c o n s i s t i n g o f q u a t e r n i o n m a t r i c e s X o f o r d e r n such

-

that I

Xx'

> U, t h e c h a r a c t e r i s t i c m a n i f o l d o f w h i c h i s t h e u n i t a r y syrnplec-

t i c y r o u p IJSP(2n).

As m e n t i o n e d above, c o n s i d e r i n y t h e a n a l y t i c automorphism

y r o u p on t h e domain I

- XX'

> 0, we can o b t a i n t h e c o r r e s p o n d i n g P o i s s o n

kernels (1.31) where 0 < r < 1 and U

B

USP(Zn), by u s i n y t h e t h e o r y o f harmonic f - u n c t i o n s on

t h e q u a t e r n i o n domain. E m p l o y i n g t h e y e n e r a t i n y t u n c t i o n inethods used by Zhong J i a Q i n g i n t h e p r o o f o f Theoren 1.4 l e a d s t o t h e f o l l o w i n y .

THEOREM 1.5.

(He Zu Qi and Chen Guang Xiao, see ( 1 1 ) .

I n t h e expansion o f

P o i s s o n k e r n e l s on u n i t a r y s y i n p l e c t i c g r o u p s P(rI,U)

=

$ pf(r)N(f)xf(U).

t h e c o e f f i c i e n t s have t h e e x p r e s s i o n

1

where

5 (r) 1

.

r

fl+2n

..... t n ( r )

.

r

S p )

f ,+n+l

,~

n n-1 n+l + r ~ + ~ =( rr ,) ~ , + ~ ( r =) r

Harmonic Analysis on Classical Groups

77

A s i n t h e case o f u n i t a r y y r o u p s , we a r e a b l e t o s t u d y t h e Abel summation o f F o u r i e r s e r i e s on r o t a t i o n o r u n i t a r y s y m p l e c t i c y r o u p s , and on t h e h a s i s o f Theorems 1.4 and 1.5 we o b t a i n t h e f o l l o w i n y c o r r e s p o n d i n y r e s u l t : The F o u r i e r s e r i e s o f any c o n t i n u o u s f u n c t i o n on t h e l a t t e r t w o c l a s s i c a l y r o u p s i s a l w a y s Abel-summable t o i t s e l f . 2. The Cessaro Sumnation The s e r i e s o f methods e s t a b l i s h e d i n t h e r e s e a r c h f o r t h e h a r m o n i c a n a l y s i s on u n i t a r y y r o u p s a r e w i d e l y a p p l i e d t o t h e r e s e a r c h f o r t h e h a r m o n i c a n a l y s i s on c l a s s i c a l y r o u p s and on compact L i e y r o u p s .

F o r example, t h e methods " f r o m

sums t o k e r n e l s " and " f r o m k e r n e l s t o sums" t o b e i n t r o d u c e d i n t h i s s e c t i o n j u s t come f r o m t h e i d e a s used i n t h e r e s e a r c h f o r t h e h a r m o n i c a n a l y s i s on u n i t a r y yroups.

By a p p l y i n g t h e s e t w o methods, t h o s e r e s u l t s o b t a i n e d by u n i t a r y

y r o u p s i n t h i s s e c t i o n and t h e subsequent ones can be e s t a b l i s h e d on c l a s s i c a l y r o u p s and on compact L i e y r o u p s .

I n f a c t , as f a r as we know, t h e s e t w o

methods a r e a l m o s t a p p l i c a b l e t o v a r i o u s t y p e s o f summation, c e n t r a l o p e r a t o r s and c e n t r a l m u l t i p l i e r s e s t a b l i s h e d on c l a s s i c a l y r o u p s and on compact L i e y r o u p s a t home and abroad. The summation c o e f f i c i e n t s o f t h o s e summations and c e n t r a l m u l t i p l i e r s e s t a b l i s h e d by t h e method " f r o m k e r n e l s t o sums",

such as A b e l - and Cesaro-

summation i n t h i s a r t i c l e and t h e c l a s s o f c e n t r a l m u l t i p l i e r s e s t a b l i s h e d t h r o u y h t h e F o u r i e r t r a n s f o r m a t i o n f o r L i e a l y e b r a s by R . S . S t r i c h a r t z [14], a r e u s u a l l y very complicated.

1.5,

1.4,

2.7

2.3,

and 2.9

H e r e o n l y t h o s e c o e f f i c i e n t s i n Theorems 1.2,

a r e c o n c r e t e l y y i v e n and t h e i r d e t e r m i n a t i o n depends

on t h e c o m p l i c a t e d c a l c u l a t i o n and s k i l l f u l methods m e n t i o n e d above. For studyiny t h e properties o f Fourier series, C e s a r o summations i n t h i s s e c t i o n , t a r y yroups.

L e t u(U)

such as t h e c o n v e r g e n c e o f

t h e f o l l o w i n y method i s e s t a b l i s h e d on u n i -

e L ( U n ) , and qJ,(V)

= c-1

J

u"

u(uwvw-l)fi.

The method i s t o s t u d y F o u r i e r s e r i e s o f u ( U ) t h r o u y h t h e c l a s s o f f u n c t i o n s {$,,(V),

U 6 Un}.

As qJU(V) i s a c l a s s f u n c t i o n ,

F o u r i e r s e r i e s of a c l a s s of f u n c t i o n s (qJu(e where JIU(e

iel

,

..., e

ien

iel

we o n l y need s t u d y m u l t i p l e

,

..., e

iBn

) , U E Un} on t o r u s ,

) a r e t h e v a l u e s o f $u(V) a t t h e maximum t o r u s , i . e .

diayonal u n i t a r y matrices.

at

L a t e r on, t h i s rnethod was a l s o used i n t h e r e s e a r c h

f o r t h e h a r m o n i c a n a l y s i s on c l a s s i c a l y r o u p s and on compact L i e y r o u p s . r e l a t e d examples can be f o u n d i n [ l O ] e x c l u d i n y t h o s e c o n d u c t e d a t home.

The

I n s p i r e d by t h e Abel-summation on u n i t a r y g r o u p s we have d e f i n e d C e s a r o means of F o u r i e r s e r i e s (1.9)

on u n i t a r y y r o u p s i n

[el.

N o t o n l y can t h e k e r -

n e l s be e x p l i c i t l y r e p r e s e n t e d by m a t r i c e s , b u t b o t h t h e summation c o e f f i c i e n t s

S. Gong et al.

78

and t h e r e l a t e d i n t e y r a l c o n s t a n t s can be c a l c u l a t e d o u t e x p l i c i t l y .

For

u n d e r s t a n d i n g o f t h e y e n e r a l Cesaro means, F e j e r means, w h i c h i s one o f t h e most t y p i c a l and most i m p o r t a n t example o f Cesaso means, was s t u d i e d c a r e fully.

T h i s example i n d i c a t e d t h a t t h e o t h e r c o e f f i c i e n t s and c o n s t a n t s

r e l a t e d t o t h e y e n e r a l Cesaro means can be o b t a i n e d i n t h e same way. L e t u ( U ) be an i n t e y r a b l e f u n c t i o n on Un and t h e Cesaro (C,a) F o u r i e r s e r i e s (1.9)

means o f i t s

be

where

H;(N)

=

xy 1 r~/ xf(V)K;(V)i

(2.2)

un and Ka(V) i n (2.2) N

i s Cesaro (C,a)

N

1 $1;

detn[

k=O

B;=c

1

VK(I

1

N

:A:;

/

k=O

un

(2A;)n Aa =

and

N

-

V'2kf1)j (2.3)

detn where

kernel which i s equal t o

b+a+)

Vk(I

2

detn(I

... ( a + l )

N!

- V'2kt1) -

v')

0,

,

moreover, a > -1 i s needed. F o r Cesaro means on u n i t a r y y r o u p s we have

THEOREM 2.1 C3l.

Cesaro (C,a)

means (2.1) o f F o u r i e r s e r i e s ( l . Y )

o f any

i n t e y r a b l e f u n c t i o n u(U) on u n i t a r y y r o u p s can be e x p r e s s e d as

1 C /

u(V'U)K;(V)t.

(2.5)

'n

PROOF.

From (2.2)

T h e r e f o r e (2.5)

and ( 2 . 3 ) ,

we have

can be w r i t t e n as

w h i c h i s j u s t t h e (2.1). I t can be i m m e d i a t e l y seen t h a t t h e r e l a t i o n between (C,a) k e r n e l s o f Cesaro

(C,a)

means d e f i n e d as above and P o i s s o n k e r n e l s o f Abel means d e f i n e d by Hua

Hurmoriic Anul-vsis or1 Classical Groups i s t h e same as i n t h e case o f F o u r i e r s e r i e s ,

i.e,

(C,a)

19

k e r n e l s become P o i s s o n

kernels i t a tends t o i n f i n i t y .

c31 L e t u ( U ) be c o n t i n u o u s on (In. Then, when

THEOREM 2.2. s e r i e s (1.9)

o f u ( U ) i s (C,a)

(O <

class L i p P

P < I),

lu(u)

-

~ : ( u ) I < A ~ N - ~ ,i f

2)

lu(u)

-

IF;(U)I

<

3)

lu(u)

-

I;(IJ)I

< A ~ N - ~ " + ~ - ~i f,

PROOF.

Fourier

satisfies

1)

where A1,

> (n-l)/n,

suinmable t o i t s e l f , and when u ( U ) b e l o n y s t o

I ~ ( u )( s e e ( 2 . 1 ) )

A

a

~

N

-

a n - n + l > P;

~ ri,~

Oi

f~ a n - n + l =

P;

an-n+l < P ;

A2 and A3 a r e a b s o l u t e c o n s t a n t s . Take

n

= max{s-(atl)(n-l),

Cesaro k e r n e l s K i ( B ) ,

O}.

1 k;(e) 1 ('-') I e I Take s = 2 ( n - 1 )

From an e s t i m a t e o f o n e - d i m e n s i o n a l

we can o b t a i n

i n the definition of

0,

< BNn-l-S

I NO 1'.

and

then

The d i r e c t e s t i m a t e o f ( 2 . 7 )

leads t o t h e conclusion.

Amony Cesaro k e r n e l s , t h e k e r n e l o f F e j e r means has t h e s i m p l e s t e x p r e s s i o n

BN( N+1) I'

where BN i s a number such t h a t t h e i n e y r a l o f (2.8) Bf(N)

=

1

Bf(N) =

1

on U,,

i s e q u a l t o 1 and

-7

N ( f ) c BN(N+l)

F e j e r means o f F o u r i e r s e r i e s (1.9

nN>fl>

1 ... >fn>-nN

o f u(U) reads

Bf(N)N(f)tr(CfAf(U))

The f o l l o w i n y t h e o r e m y i v e s t h e F e j e r means c o e f f i c i e n t s and t h e i n t e y r a l c o n s t a n t s o f F o u r i e r s e r i e s on u n i t a r y g r o u p s .

I n t h e p r o o f o f t h e theorem, a

c o m p l i c a t e d and i n g e n i o u s s k i l l f o r m a t r i x i n t e g r a t i o n i s used.

The same

S. Gong e t al,

80

lnethod can a l s o be a p p l i e d t o t h e c a l c u l a t i o n o f t h e c o e f f i c i e n t s and t h e i n t e y r a l c o n s t a n t s f o r g e n e r a l (C.a) THEOREM 2.3.

means.

[Sl The F e j e r means c o e f f i c i e n t s o f F o u r i e r s e r i e s on u n i t a r y

groups a r e Bf(N) =

2n x

1

sl=o kl>O

(-l)n(n-l)/2(lto(l/N))

2n n+k ntk, ...sn=o 1 Cgn Cn ... C f n Cn 1 n

N((N+l)sl-fl

,...,( N + l ) s n - f n ) ,

(2.10)

kn>O

where k . = f . + n - j t n N - ( N + l ) s j , J J equal t o

j = 1,2,

...,n,

and t h e i n t e y r a l c o n s t a n t s RN a r e

(2.11) From t h e d e f i n i t i o n o f N ( f ) and Theorem 2 . 3 , Bf = 1 Here we s k e t c h t h e p r o o f .

+

i t can be seen t h a t

0(1/N).

F o r d e t a i l s , see [3].

Let (2.12) Then i f p c 2n, we have

k+(N+l)s=q

-

-

(2n)!

c

(p+q-( N + l ) s - l ) ! (q-(N+l)s)!s!(Zn-s)!

2n s=o

(2.13)

q- ( N+1) s>O and tltnN

Bf(N)N(f) = (N+l)

-n2

(-1)

n(n-1)/2

-1 BN

a2"

t2+nN aZn

tn+nN

tl+nN '2n-1

t2+nN a2n-l ).***

,..., a2n

tn+nN a2n-l

.............. tltnN a

n+l

t2tnN a n+l

tn+nN

,..., an + l

(2.14)

Harmonic Analysis on Classical Groups

81

( 2n+kl-1 ) !

(2n+kn-1)!

""'

kl!s1!(2n-sl)!

n

kl >U

BN'( ( 2 n ) !)n( - 1 ) n ( n - 1 ) / 2 ( 2 n - 1 ) ! (2n-2)!

kn>O

2n

...n! (N+1 )n2

k1>0

(2n-l+kl)!

2n n kn>U

.....................

2n

sl=u 1

2n

(n+l+kl-l)! k l ! s p n - s p

.

.

kn>U

.... kn .an+nN-(N+l)sn;

'llfnN-(N+l)sl,

.... 1' ,

Ek = f k + n - k ,

f,.

S i m p l i f y i n y (2.15)

.

B

n

(0,O

.....0 ) .

2

(N+I)"

)n(n-l)!

....

fl+n-1,

2n ... 1 (-1) s =o s =o

1

Sl+.

n kn>U

where k . = n - j + n N - ( N + l ) s . J J

... 2 ! 1 !

... n !

(zn-l)!

2n

1 kl>U

.

f u r t h e r and a p p l y i n y i n g e n i o u s

Thus

= (-l)n('-')/'(n!

x

al

(2.15)

E s p e c i a l l y , we have H f ( N ) = 1 if

c a l c u l a t i o n e a s i l y l e a d t o (2.1U). f

(n+l+kn-l)!

k !s ! ( 2 n - s n ) ! sn=O n n

"*"

kl>U

where kl

(2n-l+kn-1)!

..+sn

.

'gn

n+kl

1

(N+l)(n-s.)-j, J

... n:C j

n

n+kn Cn N((N+l)sl

.l , Z , . . . , n sj .

. ....( n - 1 )

I t i s known f r o m t h e d e f i n i t i o n o f k j t h a t U,l, f o r k j > 0 i f N > n-1. The u s u a l method b e i n y used [lJ, (2.16)

.....( N + l ) s n ) (2.16)

i s necessary becomes

A s e r i e s of s k i l l f u l c a l c u l a t i o n r e l a t e d t o (2.17) h a v i n y been made [3], (2.11)

i s obtained.

Generally, series i s summation

l e t u(0) be an i n t e g r a b l e f u n c t i o n on 0 <

lm a p=-"

P

eipe.

e < 2n

and i t s F o u r i e r

Suppose t h a t T i s a summation and t h e k e r n e l s o f t h e

S. Gong et a1

82

are (2.18) N a t u r a l l y , t h e r e i s an a s s u m p t i o n o f t h e e x i s t e n c e o f t h e k e r n e l k m ( e ) , 7.e.

If

t h e converyence o f (2.18).

I;=a, eipe P

s f o r m tending t o a l i m i t , then

T,,, +

i s c a l l e d T-summable t o s.

L e t u ( U ) be an i n t e y r a b l e f u n c t i o n on \In arid i t s F o u r i e r s e r i e s be

1 N(f)tr(CtAf(U)).

(2.1Y)

f

Ayain l e t -1

= 6,

T,,,(V)

detn

(

umkVk)

-m

(2.20) Then t h e T-means o f (2.19)

is

F

(2.21)

Bf(m)N(f)tr(CfAf(U)),

where

(2.22)

B

and e

iel

,...,e i o n

m

.

i n (2.2U)

= c-l

j'

detn (

"n

(2.23)

pmkVk)v,

-m

a r e t h e c h a r a c t e r i s t i c r o o t s o f V.

Generally, replaciny km(81)

...kI,,(en)

i n (2.20)

by k e r n e l s kln(el,e2,,..,en)

o f m u l t i p l e F o u r i e r s e r i e s , we can y i v e a summation on u n i t a r y y r o u p s .

This i s

t h e method from k e r n e l t o sum.

THEOREM 2.4.

c11

L e t k e r n e l s k m ( e ) i n (2.20)

f o r any y i v e n 6 > 0 , where 6 c / e l c

1)

k N ( B ) = O(N-n)

2)

k N ( e ) = O(Nc) f o r any

3)

j'

(TN(V)(v c

satisfy

H, (m

e where 1 > 6 > 0 and

= 1,2,

...,n ) ,

where H,

TI,

> 6, a r e c o n s t a n t s dependent o n

"m

m only. Then t h e T-means (2.21)

o f F o u r i e r s e r i e s o f u(U) converges t o u ( U ) i f u(U)

i s c o n t i n u o u s on UnF o r t h e summation o f F o u r i e r s e r i e s on u n i t a r y y r o u p s s e t up by t h e method " f r o m sum t o k e r n e l " , we may b e y i n w i t h y i v i n y a c o r r e s p o n d i n y sum o f F o u r i e r s e r i e s on u n i t a r y y r o u p s by (2.18)

[l]

Harmonic Analysis on Classical Groups T,(u)

where e

f

,

( U 1)

(2.24)

..., e

(2.25)

ien

a r e t h e c h a r a c t e r i s t i c r o o t s o f V, D(x1,x2

Obviously,

1t r ( C f A

o f T-summation o f t y p e I 1 a r e

The k e r n e l s T;(V)

iel

Uman N ( f

2

T-summable t o s o f t y p e I 1 i f r m ( U ) + s when m t e n d s t o a

and c a l l (2.19)

limit.

1 lJmelUmk

=

83

, . . . I

Xn) =

n

l t i < jt n

(Xi

-

and

x.). J

i f we t a k e

... vmLn

as t h e summation k e r n e l s o f m u l t i p l e F o u r i e r s e r i e s and r e w r i t e i n (2.24)

,...,‘

, t h e n we d e f i n e a summation on u n i t a r y g r o u p s , t h e

as

1 n k e r n e l s o f w h i c h can be o b t a i n e d by c h a n y i n y km(el) km(e1,e2,..

.,en).

...k m ( e n )

i n (2.25)

into

and t h e n t h e Abel summation o f t y p e I 1 i n (2.1Y)

Take urp = 1-1’1,

i s g i v e n ( s e e C21). Choose uNk = A!(N) C e s a r o (C,a)

= A!

= r ( a + N - l k ( + l ) r ( N + l ) / ( r ( a + N + l ) r ( N - J k ( +and l)) then

summation o f t y p e 11 T,(U)

=

N>el>.

I:

..>en>-N

A’

‘1

... A:

n

N(f)tr(CfAf(U))

(2.27)

i s y i v e n [see C21). The k e r n e l (2.25)

c o r r e s p o n d i n g t o summations (2.26)

and (2.27)

t a k e s one-

d i m e n s i o n a l P o i s s o n k e r n e l and o n e - d i m e n s i o n a l Cesaro k e r n e l r e s p e c t i v e l y as k, ( 0 ) r e s p e c t iv e l y

.

F o r Abel and Cesaro summation o f t y p e 11, t h e f o l l o w i n y t h e o r e m i s v a l i d .

THEOREM 2.5.

[2]

L e t u(U) be a f u n c t i o n h a v i n g c o n t i n u o u s p a r t i a l d e r i v a t i v e s

up t o o r d e r n ( n - 1 ) / 2 , converges t o u ( U).

t h e n t h e A b e l - o r Cesaro-summation o f t y p e I 1 u n i f o r m l y

S. Gong et al.

84

Many Shi Kun and Dony Dao Zheny d e f i n e d Cesaro k e r n e l s on r o t a t i o n groups

SO(n) K:(r) where

r

8

N

+ 1

= (B:)-ldet((A:I

S O ( n ) and B:

j =1

(rJ+r'J)

N-j

1

r=U

A:-1)/A:)n(n-1)'2,

(2.28)

on S O ( n ) i s

i s a number such t h a t t h e i n t e y r a l o f K;(r)

equal t o 1.

If' u ( r ) i s i n t e y r a b l e on S O ( n ) , i t s F o u r i e r s e r i e s i s

u(r) where

h(r) a r e

m = (ml m

,...,mk)

> m2 > 1

- m1

N(m)tr(CmAm(r)),

(2.29)

t h e i r r e d u c i b l e r e p r e s e n t a t i o n s o f SU(n) w h i c h t a k e

...

> > mk > U a r e i n t e y e r s i f n = 2 k + l and > l m k ( > U a r e a l s o i n t e y e r s i f n = 2k, N(m) = N(ml

as i t s l a b e l s , ml

... > mk-l

,...,

m k ) i s t h e o r d e r o f A m ( r ) , and

where c i s t h e volume o f SO(n) and L e t Xm(r) = tr(A,(r))

?

i s t h e volume e l e m e n t .

and (2.31)

I t i s e a s i l y seen t h a t

i f n = 2 k , m1 > coefficients

... > mk

B i 1,. ..,m

Cesaro (C,a)

k

> 0. T h e r e f o r e , we o n l y need t o c a l c u l a t e t h e f o r ml > > mk > U.

...

means o f (2.29

are (2.32)

THEOREM 2.6.

(Wany Shi Kun and Dong Dao Zheny, see [ l ] )

t i n u o u s f u n c t i o n on SU(n), s e r i e s (2.29)

of

u(r)

r

6

i s (C,a)-summable

t o i t s e l f and, when

1)

JCi(r) -

u ( r ) J < A ~ N - P ; i f a(n-1)tz-n

2)

ll:(r)

u(r)(

3)

IC;(r) -

L e t u ( r ) be a con-

S o ( n ) , t h e n , when a > ( n - Z ) / ( n - l ) ,

< A ~ N - P ~ ON,Y

,

the Fourier L i p p,

> p;

i f =(n-1)+2-n

n-2-a( n-1) u(r)) < A ~ N

u(r) e

= p;

i f a ( n - l ) + Z - n < p.

When a = 1, t h e Cesaro summation i s j u s t t h e F e j e r summation and i t s k e r n e l s are

Harmonic Analysis on Classical Groups

KN ( r )

=-

1 BN

lihen n = Zk, (2.33)

I

N

. N-j+l

1 r-' N+l

det(1 + 2

j =1

85

I (n-1)/2

)

(2.33)

becomes

and

THEOREM 2.7.

(liany Shi Kun and nony Dao Zheny, see [ I ] )

On S 0 ( 2 k ) , t h e F e j e r

summation c o e f f i c i e n t s r e a d

n

((2k-1)!)k B ml..

.m k

N( i n ) ( 2 k - 1 ) !

...

Sk'0

4k-2

(-1)

sl+ ...+ s

1 el>l-k

3k-2+el

k '4k-2 s k

...

1=u

...( 4 k - 3 ) !BN(N+1) k ( 2 k - 1 )

4k-2

1

(s2-j2)

O
-

'2k-1

".

'4k-2 sk

3k-2+ek '2k-1

*"

ek>l-k

... N( ( n - l - s l

) ( N+l)-ml,

where e. J ( 2 k - l ) N - ( N + l ) s . - m . - k + j , 3 3 3 i n t e y r a l constants are ((2k-l)!)k B

=

(Zk-l)!(Zk+l)!

..., ( n - l - s k )

j = 1,2

,...,k

and m l >

O < jn< s < k - l ( s 2 - j 2 )

... ( 4 k - 3 ) ! ( N + l ) k ( 2 k - 1 )

where e j = ( P k - l ) N - ( N + l ) S j - k + j ,

j = 1,2

( N+1 )-mk),

,...,k.

On S 0 ( 2 k + l ) t h e F e j e r summation c o e f f i c i e n t s a r e

4k-2

1=o

1 el>l-k

... > mk > U and

...

its

86

S. Gong et al. ((2k)!)k

B,

=

;...

(k

- );

n

...

N(m) ( 2 k ) ! ( 2 k + 2 ) !

... Sk'04k1

4k x

1

sl=o el>l-k k

x

((2k)!)k N

=

-

4k

1

s 1=u

el>l-k

1

j = 1,2,..

J

J

-!j- ... ( k -

1

,...,( n - 1 - s k ) ( N + l ) - m k ) ,

-

1'

((s +

II

.,k

and t h e i n t e y r a l

( j +;

)2)

U
... ( 4 k - 2 ) ! ( N + l ) k ( 2 k - 1 )

... s I:=o (-1) 4k

Sl+.

..+s k C4k

4k

sl"'csk

3k-l+el '2k

3k-l+e

...Cilk

k

k ek>l-k

k

x

...C 4ksk '2k3k-l+el ...C2k3 k - l + e k

) - l N ( ( n-l-sl)(N+l)-ml

1 . = m.+k-j,

(2k)! (2k+2)!

x

C4k s1

-1)

ek>l-k

where e . = Z k N - ( N + l ) s . - k . , J J J constants are

B

.+S

i)2)

+

(4k-2) !BN(N+l) k ( 2 k

.

sl+. (-1)

((n-1-s. ) ( N+l)-L. J J

lI

j =1

( ( sj (-2)+- j !-

U
II ( ( n-1-s. ) ( N + l ) - k + j -

j =1

)-l N( ( n - l - s l ) (

J

where e j = P k N - ( N + l ) s j - k + j , The p r o o f o f Theorem 2.7

j = 1,2

N+1),

...,( n - l - s k ) (

N+l))

,

,...,k.

needs t h e method used i n Theorem 2.3

c o m p l i c a t e d and s k i l l f u l c a l c u l a t i o n .

For d e t a i l s ,

see

and needs a

El].

He Zhu Qi and Chen Guang X i a o d e f i n e d Ceasro k e r n e l s and Cesaro summation on u n i t a r y s y m p l e c t i c groups USP(2n), n = 1,2,

... .

L e t u ( U ) be i n t e g r a b l e on USP(2n) and i t s F o u r i e r s e r i e s i s u(U) where f = ( f l , f2

,...,f n ) ,

fl

- 1f

> f2 >

N(f)tr(CfAf(U

... > f n

u

1

(2.34)

a r e i n t e y e r s , Af(U) a r e t h e

u n i t a r y s i n y l e - v a l u e d i r r e d u c i b l e r e p r e s e n t a t i o n s o f USP(2n) w h i c h t a k e f as i t s l a b e l s , N ( f ) are t h e orders o f Af(U), x f ( U ) = t r ( A f ( U ) ) a r e t h e characters o f Af(U),

and

cf

= c-1

J

USP(2n)

where c i s t h e volume o f USP(2n) and So, Cesaro means o f (2.34)

is

6

u ( U)Af

(g')I?,

i s t h e volume element.

Harmonic Analysis on Classical Groups

87

(2.35)

where

(2.36)

i s t h e Ceasro (C,a)

kernel.

I

USP( 2 n )

f o r N = 1,2,3,

...

THEOREM 2.8.

(He Zu

B E a r e t h o s e numbers such t h a t

I n (2.37),

Ki(V);

= 1

C il and Chen Guany Xiao, see [ l ] )

on USP(2n), U 6 USP(Zn), t h e n when a > ( 2 n - 2 ) / ( 2 n + l ) , o f u ( U ) i s (C,a)-summmable

IT;(U)

t o i t s e l f , and when u ( U )

- U(IJ)( < I~;(u) - u ( u ) ( < l r ~ ( ~- ) u ( ~ )
1) 2) 3)

L e t u ( U ) be c o n t i n u o u s t h e F o u r i e r s e r i e s (2.34)

E

L i p p, t h e f o l l o w i n y h o l d

A ~ N - P , i f ( ~ n + l ) a - ~ n +>2 p;

A ~ N - P l o g N, i f ( 2 n + l ) a - ~ n + 2 = p; A ~ N ~ ~ - ~ - i(f ~( ~~n ++l ) a~- 2) n +~ 2 <, p.

I f a = 1, i t i s j u s t t h e F e j e r summation and i t s F e j e r k e r n e l s a r e 1

BN( N+1)n(2n+1) (

-

det(1 VN+l) )2n+l d e t ( 1 - V)

S i m i l a r t o t h e p r o o f o f Theorem 2.3 we can o b t a i n t h e F e j e r summation c o e f f i c i e n t s and i t s i n t e y r a l c o n s t a n t s .

THEOREM 2.9.

(He Z u Qi and Chen Guany Xiao, see 1 1 1 )

The F e j e r summation

c o e f f i c i e n t s o f F o u r i e r s e r i e s on t h e u n i t a r y s y m p l e c t i c y r o u p s a r e t h e followiny

x

4n+2

1

sl=o

... 4n+2 1 sn=o

kl>l-n 3n+kn 'kn+n-l

(-1)

S1+"'+Sn

C4n+2 s1

3n+kl ...c4n+2 Ck+n-l ... 'n

kn>l-n

N( (n+( 2 n + l ) N - ( N+l)sl-fl,.

where k . = (2n+l)N-(N+l)sj-(fj+n-j+l), J

.., Z n - l + ( 2 n + l

j = l,Z,.

..,n,

and

)N-(N+l )sn-fn),

S. Gong et al.

88

4n+2

4n+2

3n+kn

3n+kn

‘kl+n-l kl>l-n x

“ ’ ckn+n-l(-l)

n

Sl+...+S

kn>l-n

,..., 2 n - 1 + ( 2 n + l ) N - ( N + l ) s n ) .

N(n+(2n+l)N-(N+l)sl

L i Shi X i o n y and Zheny Xue An d e f i n e d and d i s c u s s e d Ceasro k e r n e l s and Cesaro summation o f F o u r i e r s e r i e s c o n n e c t e d w i t h compact L i e y r o u p s . w i t h , Cesaro (C,a)

a l g e b r a i s one o f t h e compact L i e a l g e b r a s (A,,)U, (F4lu,

F6

(E6)u,

TO b e g i n

k e r n e l s a r e d e f i n e d on any compact L i e groups whose L i e

( E 7 l u y ( E 8 ) u and un = ( A n - l ) u

@

(Bn)U, (Cn)u,

H1,

(G2)u,

6B H1,

Y2 =

e6 = (Efj)u f3

= ( E s ) u @I H2,

e7 = (E7),, 6B H1 and Hn w h i c h i s t h e L i e a l g e b r a o f t o r u s

Tn w i t h d i m e n s i o n n.

These L i e a l y e b r a s a r e u s u a l l y c a l l e d t h e b a s i c compact

Hl,

L i e alyebras.

F o r a g e n e r a l compact L i e g r o u p G, t h e L i e a l g e b r a o f I; can be

decomposed e i t h e r as a d i r e c t sum w h i c h c o n s i s t s o f t h e b a s i c compact L i e a l g e b r a s l i s t e d above e x c e p t (An)U, (G2)u,

(E6)u,

e6,

( E 7 ) u , o r as a d i r e c t sum

w h i c h c o n s i s t s of t h e b a s i c compact L i e a l y e b r a s l i s t e d above e x c e p t Hn and a t l e a s t one o f (An),,,

( G Z ) ~ , (E6)u, e6, (E7),,i s i n c l u d e d i n i t .

t h e r e y u l a r d e c o m p o s i t i o n o f a compact L i e a l g e b r a .

This i s called

Here t h e Cesaro k e r n e l o f

G i s j u s t a p r o d u c t o r some r e s t r i c t i o n o f t h e p r o d u c t o f Cesaro k e r n e l s o f s e v e r a l b a s i c compact L i e groups m e n t i o n e d above, w h i c h c o r r e s p o n d t o t h e r e g u l a r d e c o m p o s i t i o n o f t h e L i e a l g e b r a o f G.

THEOREM 2.10. ( L i Shi X i o n y and Zheny Xue An) group. whose L i e a l g e b r a i s one of (An)u, (Es),,,

(E7),,,

.-,

,

( E B ) ~ , un, 92, e6, e6

We h a v e ( 1 ) L e t G be a compact L i e

(Bn)us ( C n ) u , (Dn)u, (Gillu,

(F4Iu,

e7, and a g a i n l e t t h e c r i t i c a l v a l u e s a.

c o r r e s p o n d i n y t o t h e above-mentioned b a s i c compact L i e a1 gebras b e

respectively.

Then t h e Cesaro means

o f F o u r i e r s e r i e s of any c o n t i n u o u s f u n c t i o n f ( x ) on G u n i f o r m l y c o n v e r g e s t o f ( x ) i f a > ao, where x denotes convolution; satisfy

6

G, K’(x)

n

s t a n d f o r Cesaro ( C , a ) k e r n e l s on G,

and if f ( x ) b e l o n y s t o L i p p and . a

= a/b,

*

t h e n T;(x)

Harmonic Analysis on Classical Groups

d)

Iri(X)

-

f ( X ) ) < AIN-P,

b)

IT:(x)

-

f ( x ) ( < A2N-P

C)

IT;(x)

-

f(x)

I

89

i f ab-a > p;

l o g N, i f ab-a = p;

< A3Na-ab,

i f ab-a < p;

where a and b a r e g i v e n i n ( 2 . 3 8 ) . 2)

L e t G t a k e L as i t s L i e a l g e b r a and t h e r e y u l a r d e c o m p o s i t i o n o f L be L = L1 f3 L 2 f3

... f3 L k .

By a o ( L j ) we d e n o t e t h e c r i t i c a l v a l u e s c o r r e s p o n d i n g t o L j , j = 1,2,...,k,

and

set

Then t h e Cesaro summation o f F o u r i e r s e r i e s o f any c o n t i n u o u s f u n c t i o n f ( x ) on

G u n i f o r m l y c o n v e r g e s t o f ( x ) i f a > ag.

Moreover, i f a0 = a/b,

then a), b),

and c ) c o r r e s p o n d i n g t o 1 ) a r e a l s o v a l i d . The T-summation and T-summation o f t y p e I 1 o f F o u r i e r s e r i e s on u n i t a r y g r o u p s e s t a b l i s h e d by t h e methods " f r o m k e r n e l t o sum and f r o m sum t o k e r n e l " have s i m i l a r e x t e n s i o n s on compact L i e g r o u p s .

The r e l a t e d d e t a i l i s o m i t t e d .

3. The Cubical Partial Sums o f Fourier Series I n t h i s s e c t i o n we c o n s i d e r b r i e f l y t h e d e f i n i t i o n o f t h e c u b i c a l p a r t i a l sums o f F o u r i e r s e r i e s on u n i t a r y g r o u p s and i t s e x t e n s i o n s on c l a s s c a l g r o u p s and on compact L i e y r o u p s . I n t h e p r o o f o f Theorem 3.1,

i n which t h e concrete expression f o r D i r i c h l e t

k e r n e l s o f t h e c u b i c a l p a r t i a l sums o f F o u r i e r s e r i e s on u n i t a r y y r o u p s was e s t a b l i s h e d , a b a s i c method o f s t u d y i n y c l a s s f u n c t i o n s was s e t up.

The

essence o f t h e method i s t o t r a n s f o r m a r e s e a r c h p r o b l e m on c l a s s f u n c t i o n s t o a p r o b l e m on F o u r i e r s e r i e s o f t h e f u n c t i o n s ( s u c h as g(el,

...,en)

i n (3.6))

on

t o r u s w h i c h a r e made by t h e p r o d u c t o f v a l u e s a t t h e maximum t o r u s o f c l a s s f u n c t i o n s and t h e Weyl f u n c t i o n .

T h i s method i s a l s o w i d e l y a p p l i e d t o

r e s e a r c h f o r c l a s s f u n c t i o n s on c l a s s i c a l g r o u p s and compact L i e groups.

Some

r e s e a r c h e r s a b r o a d such as R. J . S t a n t o n and P. A. Tomas a d o p t e d t h i s method i n t h e i r studies

on t h e a l m o s t e v e r y w h e r e c o n v e r g e n c e o f F o u r i e r s e r i e s o f c l a s s

f u n c t i o n s on compact L i e y r o u p s . The c u b i c a l p a r t i a l sum o f F o u r i e r s e r i e s on u n i t a r y g r o u p s have t w o f o r m s

o f e x t e n s i o n s on compact L i e g r o u p s .

One i s made by R. J. S t a n t o n and P. A.

Tomas They, s t a r t i n g f r o m t h e convex p o l y h e d r o n ( i n c l u d i n g t h e o r i g i n as i t s i n t e r i o r p o i n t ) on C a r t a n sub-a1 g e b r a s o f L i e a1 g e b r a s o f compact L i e g r o u p s w h i c h i s i n v a r i a n t u n d e r Weyl g r o u p s , d e f i n e d t h e p o l y h e d r a l p a r t i a l sums, f o r

S. Gong e l al.

90

w h i c h one o f t h e fundamental p r o p e r t i e s f o r t h e c u b i c a l p a r t i a l sums d e f i n e d on u n i t a r y g r o u p s was used.

A n o t h e r i s made by L i S h i X i o n g and Zheng Xue An.

They, s t a r t i n g f r o m t h e r e g u l a r c o o r d i n a t e s f o r t h e h i g h e s t w e i g h t s i n a cube o r a p o l y h e d r o n , d e f i n e d t h e c u b i c a l and p o l y h e d r a l sums o f F o u r i e r s e r i e s on compact L i e groups, f o r w h i c h a n o t h e r b a s i c p r o p e r t y f o r t h e c u b i c a l p a r t i a l

sums d e f i n e d on u n i t a r y groups was used. F o r e x p r e s s i n g D i r i c h l e t k e r n e l s e x p l i c i t l y , a d i f f e r e n t i a l o p e r a t o r was e s t a b l i s h e d on u n i t a r y groups, by means o f w h i c h D i r i c h l e t k e r n e l s on u n i t a r y groups c o u l d be s i m p l y e x p r e s s e d by D i r i c h l e t k e r n e l s o f m u l t i p l e F o u r i e r series.

Moreover, when we e s t a b l i s h T-summation k e r n e l s o f t y p e I 1 i n s e c t i o n

I 1 and when we deduce t h e i n t e g r a l r e p r e s e n t a t i o n s o f t h e s p h e r i c a l means summation, t h i s o p e r a t o r a l s o p l a y an i m p o r t a n t r o l e .

Wany Shi Kun, Dony Uao

Zheng, He Zhu Qi, Chen Guang X i a o e s t a b l i s h e d c o r r e s p o n d i n g d i f f e r e n t i a l o p e r a t o r s on r o t a t i o n y r o u p s and u n i t a r y s y m p l e c t i c groups r e s p e c t i v e l y .

Li

S h i Xiong and Zheng Xue An e s t a b l i s h e d c o r r e s p o n d i n g d i f f e r e n t i a l o p e r a t o r s on g e n e r a l compact L i e groups and gave t h e i r r e p r e s e n t a t i o n s under v a r i o u s systems o f coordinates e x p l i c i t l y . Some r e s e a r c h e r s abroad such as J . L. C l e r c [ l l J e s t a b l i s h e d c o r r e s p o n d i n g d i f f e r e n t i a l o p e r a t o r s on ( s e m i - s i m p l e ) compact L i e g r o u p s w h i c h were e x p r e s s e d as d i r e c t i o n a l d e r i v a t i v e s . When d i s c u s s i n g t h e p r o b l e m a b o u t t h e c e n t r a l m u l t i p l i e r on compact L i e groups, K. Coifman, G. Weiss [ l o ] and N. J. Weiss [15] e s t a b l i s h e d t h e d i f f e r e n c e o p e r a t o r s s i m i l a r t o t h e d i f f e r e n t i a l o p e r a t o r s on u n i t a r y yroups. The c u b i c a l p a r t i a l sums o f F o u r i e r s e r i e s (1.9)

o f an i n t e g r a b l e f u n c t i o n

u(U) on t h e u n i t a r y group Un a r e d e f i n e d by

where L~ = fl+n-l,

k2 = f2tn-2,

..., in = fn.

Let

SN(U,U) = U * U N ( U ) = c-1

then

J

U(UV')UN(V)i,

'n

p ( V ) i s c a l l e d the D i r i c h l e t kernel. N

THEOREM 3.1.

121

d N ( a ) = I p = - N eipe,

Let

ia1

,

.,.,

t h e n we have

ie

n be t h e c h a r a c t e r i s t i c r o o t s o f V

e

Un,

Harmonic Analysis on Classical Groups

-

( - i )n(n-1)/2

'5,...,e l e n ) ( n - l ) ! ...l ! D ( e PROOF.

The f u n c t i o n

c .>tn>-N

N>tl>..

DN ( V ) ,

as Abel-means o f

(8 ))

d e t (din-J

(3.3)

.

( x ) = ( d / d x ) '-JdN( x )

where

91

Pf(r)N(f)xf

i s a class function,

V)

9

hence we o n l y need t o c o n s i d e r t h e

following series ( D ( e iel

,...,e i'n)l-l N>tl>

From t h e d e f i n i t i o n (1.8)

i S

iel

,...,e i e n ) ,

z ...

pf(r)N(f)Hf(e

of pf(r),

i t i s easy t o see t h a t t h e s e r i e s i n

>tn>-N

(3.4)

t h e c u b i c a l p a r t i a l sums o f t h e m u l t i p l e F o u r i e r s e r i e s o f t h e f u n c t i o n

y(el

,...,8,)

Thus (3.5)

= l(1-re

iel

2

i8, )...(l-re

) ) - 2 n ( l - r 2 ) n D(e

iel

,...,e

ie,

).

(3.6)

can be e x p r e s s e d as

(3.7)

I n v i r t u e o f t h e skew-symmetry o f g($l,...,$n)

,...,$,)

+ ($j

(n!)-1(2n)-n

,...,$jn ), 1 2n

I ... J

0

0

(3.5)

2n g(Q1

under t h e permutation

can a l s o be e x p r e s s e d as

,...,$n)P($l ,...,$n;

81,...yen)d$l...d$n

S. Gong et al.

92

c l a s s f u n c t i o n and i t s v a l u e f o r d i a g o n a l m a t r i c e s i s

By Theorem 1.1 t h e v a l u e o f (3.8)

$1

,...,$,

a t p o i n t $1 =

... = $n

i s t h a t o f t h e continuous f u n c t i o n o f

= 0 when r + 1.

By a r e s u l t i n [l], this is

Thus t h e c o n c l u s i o n f o l l o w s f r o m t a k i n g l i m i t i n ( 3 . 4 ) . F o r t h e u n i f o r m c o n v e r g e n c e o f t h e c u b i c a l p a r t i a l sums o f F o u r i e r s e r i e s on u n i t a r y groups, t h e f o l l o w i n g r e s u l t s a r e v a l i d . I f u(U)

[4]

THEOREM 3.2.

8

Cn(n-1)/2tp(Un)

(0 < p < l ) , t h e n t h e c u b i c a l

p a r t i a l Sums SN(U,U) o f i t s F o u r i e r s e r i e s c o n v e r g e t o u(U) and (SN(u,U) Let

u(r)

-

u(U)

1

< A max{ (N-'loyn

2

N)l/(ntl),

N-pl~gn-lN}.

be an i n t e g r a b l e f u n c t i o n on S 0 ( n ) , and t h e n t h e c u b i c a l p a r t i a l

sums o f i t s F o u r i e r s e r i e s (2.29) a r e d e f i n e d b y (3.10) i f n = 2k and

r

€

S0(2k+l);

The f o l l o w i n g lemma

s needed :

...

qk b e i n t e g e r s such t h a t q1 > q2 > > qk > 0, p j ( q s ) L e t ql... be a f u n c t i o n dependent o n l y on qs, j = 1,2 k, and l e t N be a p o s i t i v e

LEMMA 1. C8l

,...,

i n t e g e r , a and b be any r e a l numbers, t h e n

The D i r i c h l e t k e r n e l o f t h e c u b i c a l p a r t i a l sums d e f i n e d by (3.10) (3.11)

are vN(r)

therefore

"al>.

c..>en>O

N(m)om(r),

and (3.12)

93

Harmonic Analysis on Classical Groups where t h e meaniny o f a,(r)

THEOREM 3.3.

[81

Then by Lemma 1, we have

i s y i v e n i n Theorem 1.3.

I f n = 2k. then (3.13)

and i f n = 2 k + l , t h e n

where d N ( e ) = s i n ( N

...4 ! 2 ! ,

aZk = Z1-!2k-2)!

<

'

(ej)), c ( e )

det(C

1

+ ,)e1

qS

j ,s

< k , and e

=

tiel

/ sin a2k+l

0, e N ( e ) = s i n ( N + l ) e = Zmkik(2k-1)!

2 cos qe, s(ql, . . . , q k )

,...,e

fiek

...3 ! 1 ! ,

= det(S

,

/ s i n 1 8, C(q,

qS

(3.14)

,...,q k )

(ej)),

S

4

(el

are the characteristic roots o f

= = 2 i s i n qe,

r.

F o r t h e c o n v e r g e n c e o f F o u r i e r s e r i e s on S 0 ( n ) , t h e f o l l o w i n g t h e o r e m i s valid.

THEOREM 3.4.

(Wang Shi Kun and Dong Dao Zheng,

d e f i n e d on SO(n) and

u(r)

€

F o u r i e r s e r i e s (2.29) 9 r )

SO(n), moreover,

i f n = 2 k , where 0 < p

Ck(k-l)+P

1 sN ( u

r e

-

of

u(r)

converge t o

L e t u ( r ) be a f u n c t i o n see [l])

u(r)

let

6

Ck2+p i f n = 2 k + l and

1, t h e n t h e p a r t i a l sums SN(U'r)

u(r)

u ( r ) l < A max{(N - l / ( k + l ) ( l o g

Of

and

N) k 2 / ( k+l)

The c u b i c a l p a r t i a l sums of F o u r i e r s e r i e s (2.34)

,

N-P( l o g N) k-l)

1.

of integrable function

u ( U ) on USP(2n) d e f i n e d by He Zu Qi and Chen Guang X i a o a r e

where ak = f k + n - k + l . I n t h e same way, we can o b t a i n SN(U,U) = c - l

J

USP ( 2 n )

U(V'U)UN( v

)i,

(3.17)

where a r e fli r i c h l e t k e r n e l s

THEOREM 3.5.

.

(He Zu Qi and Chen Guang X i a o , see [ l ] )

L e t u ( U ) be an i n t e g r a b l e

f u n c t i o n on USP(2n), U E USP(2n), t h e n t h e p a r t i a l sums (3.15) s e r i e s (2.34)

(3.16)

can b e e x p r e s s e d as (3.16)

and

of i t s Fourier

S. Gong et al.

94

(3.18) (2n-2k+l) ! d e t ( s i n ( n-p+l)e. ) k=l J l
II

where e

tie1

,...,e

ti en

a r e t h e c h a r a c t e r i s t i c r o o t s o f V.

I t can be o b t a i n e d t h a t

(He Zu Qi and Chen Ggang Xiao, see [ l ] )

THEOREM 3.6.

f u n c t i o n on USP(2n) and u(U)

8

L

Cn 'p,

L e t u(U) be an i n t e g r a b l e

0 < p < 1, t h e n SN(u,U

c o n v e r g e s t o u( U)

u n i f o r m l y and

ISN(uyU)

-

u ( U ) ( < A max{(N-'loy n2 N) l / ( n + l ) ,

N-p(

oy N I n - ' } .

L i Shi X i o n g and Zheng Xue An s t u d i e d t h e c u b i c a l p a r t i a l sums and p o l y h e d r a l p a r t i a l sums o f F o u r i e r s e r i e s on compact L i e g r o u p s . The c o o r d i n a t e r e p r e s e n t a t i o n s o f t h e C a r t a n s u b a l g e b r a H I o f any s i m p l e compact L i e a l g e b r a a r e known.

1; H =

Aj

As t o (An-l)u,

we have H I = {hAl.,.An(

U}, t h u s t h e c o o r d i n a t e r e p r e s e n t a t i o n f o r C a r t a n s u b a l g e b r a o f un a r e

=

{ hA1.. .An}.

The c o o r d i n a t e r e p r e s e n t a t i o n f o r C a r t a n s u b a l g e b r a s o f o t h e r

b a s i c compact L i e a l g e b r a s m e n t i o n e d i n s e c t i o n 2 c a n a l s o be d e c i d e d s i m i larly.

I f L i s a compact L i e a l g e b r a , t h e n we t a k e t h e d i r e c t sums o f t h e

above-mentioned c o o r d i n a t e r e p r e s e n t a t i o n s f o r C a r t a n s u b a l g e b r a s o f t h o s e b a s i c compact L i e a l g e b r a s w h i c h a r e i n c l u d e d i n t h e r e g u l a r d e c o m p o s i t i o n o f L ( s e e s e c t i o n 2) as t h e c o o r d i n a t e r e p r e s e n t a t i o n s o f t h e C a r t a n s u b a l g e b r a o f L.

These r e p r e s e n t a t i o n s a r e c a l l e d t h e s t a n d a r d c o o r d i n a t e r e p r e s e n t a t i o n s

f o r C a r t a n s u b a l y e b r a o f compact L i e a l g e b r a s . L e t H be a C a r t a n s u b a l g e b r a o f a compact L i e a l g e b r a L, and t h e S t a n d a r d c o o r d i n a t e r e p r e s e n t a t i o n f o r t h e p o i n t s i n H be hA1...A

xl,

..., P a r e n o t A

n e c e s s a r i l y i n d e p e n d e n t o f each o t h e r .

r a l i t y , assume t h a t XlY...,A

4

P

.

G e n e r a l l y speaking,

W i t h o u t l o s s o f gene-

i s t h e maximal l i n e a r l y i n d e p e n d e n t system and

t h a t t h e system o f a f f i n e c o o r d i n a t e s composed by t h e maximal system i s c a l l e d t h e r e g u l a r system on H.

where f = (fl,

...,f 4)

Thus any w e i g h t A on H can be u n i q u e l y e x p r e s s e d as

are c a l l e d the regular coordinates f o r A.

L e t L be t h e d i r e c t sum o f a s e m i - s i m p l e compact L i e a l g e b r a L ' and t h e c e n t r e Hk.

N a t u r a l l y , as a system o f v e c t o r s on t h e C a r t a n s u b - a l g e b r a H I o f

L o , t h e system o f r o o t s o f L o i s j u s t t h e system o f v e c t o r s on t h e c o r r e s p o n d i n y C a r t a n s u b a l y e b r a H = HI tB Hk o f L w h i c h i s c a l l e d t h e system o f r o o t s

95

Harmonic Analysis on Classical Groups

f o r L and t h e g r o u p g e n e r a t e d by t h e r e f l e c t i o n s w i t h r e g a r d t o t h e r o o t s i n H i s c a l l e d t h e Weyl y r o u p f o r L. p o s i t i v e roots.

B e s i d e s , @ d e n o t e s t h e h a l f o f t h e sum o f a l l

Thus we v e r i f i e d t h a t t h e e q u i v a l e n t c l a s s o f a l l s i n y l e -

v a l u e d i r r e d u c i b l e r e p r e s e n t a t i o n s f o r a compact L i e g r o u p i s u n i q u e l y d e t e r m i n e d by t h e h i g h e s t w e i g h t s .

M o r e o v e r , we e x p l i c i t l y e s t a b l i s h t h e method o f

c a l c u l a t i n g t h e regular coordinates f o r t h e highest weights. L e t G be a c o n n e c t e d compact L i e g r o u p and ?ibe t h e s e t o f t h e h i g h e s t w e i g h t s o f a l l n o n e q u i v a l e n t s i n g l e - v a l u e d i r r e d u c i b l e r e p r e s e n t a t i o n s f o r G. Let AX(g), g f(y)

L(G).

E

E

x

which takes

G, be t h e s i n g l e - v a l u e d i r r e d u c i b l e u n i t a r y r e p r e s e n t a t i o n f o r G

as i t s h i g h e s t w e i y h t , and d X be t h e o r d e r o f A X ( g ) .

Let

The F o u r i e r s e r i e s o f f ( g ) i s u s u a l l y e x p r e s s e d as (3.19)

or where

(3.20)

c,

If X

I f(g)AX(g-’)dg

= E

G

6

IG dg

and

= 1, and X,(g)

and t h e r e g u l a r c o o r d i n a t e f o r

c u b i c a l p a r t i a l sums o f (3.19) SN(f;Y)

o r (3.20) =

N>E1,.

Assume t h a t D1(q)

N>El,.

...,E 9 )

then the

d X t r ( CXAX(Y) ) dx f*XX ( Y )

(1

= fr)N(Y)

DN(9) =

+ fl i s ( t 1 ,

are

c..,E >-N c 4 ..,E >-N

N>kl,.

where D N ( g ) i s D i r i c h l e t k e r n e l s and

x

i s t h e character o f AX(g).

9

c..,a

4

>-N

dXXX(S).

i s a p o l y h e d r o n i n E u c l i d e a n space o f d i m e n s i o n q w h i c h

t a k e s t h e o r i g i n as i t s i n t e r i o r p o i n t and t h e c o e f f i c i e n t s o f t h e e q u a t i o n s o f a l l faces being integers.

x

+ 6

E

S e t D N ( q ) = { x E E ~ , x = t y , y E D1(q),

0 < t < N).

DN(q) means t h a t t h e R e g u l a r c o o r d i n a t e s f o r A + fl b e l o n g t o UN(q).

Then t h e p o l y h e d r a l p a r t i a l sums o f F o u r i e r s e r i e s o f f ( g ) a r e

and D i r i c h l e t k e r n e l s a r e

I n t h i s c o n n e x i o n . we have t h e f o l l o w i n g r e s u l t s .

S. Gong et al.

96

THEOREM 3.7. ( L i Shi X i o n g and Zheng Xue An) L e t G be a compact L i e group, T be a maximal t o r u s o f G, dim G = n, d i m T = q, m = ( n - q ) / 2 and Lebesyue c o n s t a n t s f o r i t s U i r i c h l e t k e r n e l s be PN(G) = then

1)

AGN[n’31(10g

pN(G)

N)’,

j’ G

(’N(Y)Idy

where n I s mod 3, s = U,1,2,

[XI denotes

t h e g r e a t e s t i n t e g e r o f a l l i n t e g e r s t h a t a r e n o t g r e a t e r t h a n x, i f G t a k e s one of

2)

(Ck)u,

u k , (Bk),,, PN(G)

’

as i t s L i e a l g e b r a ;

AGN “nt1)’31(10y

Nls,where n

# 3, n + l

5

s mod 3, s = U,1,2;

if G

t a k e s ( A k ) u as i t s L i e a l g e b r a . 3)

pN(G)

5 HGN l o g N,

4)

pN(G)

AGN, i f G t a k e s one of (Al),,,

5)

pN(G) < AGNm(log N)’,

t a k e s one o f 92, (F4),,,

i f G t a k e s ( G Z ) ~as i t s L i e a l g e b r a ;

where s = 2 f o r

( E 6 l u , ( E 7 ) u , (E81u,

(Bl)u,

e6

(Cl),,

as i t s L i e a l g e b r a ;

and s = 1 f o r t h e o t h e r s , i f G

e6, e6, e7 as i t s L i e a l g e b r a ;

...

@ L i s the 6 ) PN(G) = A G P N ( G ~ ) P N ( G ~ ) pN(Gp), if L = L 1 @ L2 @ P r e g u l a r d e c o m p o s i t i o n f o r L i e a l g e b r a L o f G and Gk i s t h e b a s i c compact L i e

group o f w h i c h L i e a l g e b r a i s L k , k = 1,2 7)

,...,p;

Lebesyue c o n s t a n t o f t h e k e r n e l D;(Y)

o f t h e p o l y h e d r a l p a r t i a l sums f o r

F o u r i e r s e r i e s on G s a t i s f i e s

where s < q and s = 1 f o r t h o s e L i e a l g e b r a s i n 1 )

-

5 ) and s = 2 o n l y f o r

e6.

Moreover, t h e c o n c l u s i o n i n 6 ) i s a l s o v a l i d f o r g e n e r a l compact L i e g r o u p s .

8 ) The c o n d i t i o n f o r u n i f o r m c o n v e r g e n c e o f t h e c u b i c a l and p o l y h e d r a l p a r t i a l sums o f F o u r i e r s e r i e s and t h e e s t i m a t i o n f o r t h e a p p r o x i m a t i o n o f t h e p a r t i a l sums t o t h e f u n c t i o n s can be deduced f r o m J a c k s o n t h e o r e m ( s e e Theorem 4.14,

3)).

I n Theorem 3.7,

1)

-

d e n o t e s t h e p r i n c i p a l p a r t o f pN(G).

F o r Theorem 3.7,

4 ) , t h e e x a c t v a l u e s f o r c o n s t a n t s AG a r e a l r e a d y o b t a i n e d by us.

U s u a l l y , t h e a b s o l u t e convergence o f F o u r i e r s e r i e s on compact L i e g r o u p s i s e x p r e s s e d by (3.21)

THEOREM 3.8.

( L i Shi X i o n g and Zheng Xue An)

i n Theorem 3.7,

L e t G, T, n, q, m be d e f i n e d as

Harmonic Analysis on Classical Groups I f f ( g ) E L!'p(G),

1)

and p > n / 2

- [n/2],

U < p

and i n p a r t i c u l a r i f f ( y )

6

Ck*P(G)

91

,

where k = [ n / 2 ]

< 1, t h e n t h e F o u r i e r s e r i e s f o r f ( g ) c o n v e r g e s

a b s o l u t e l y and u n i f o r m l y , a c c o r d i n g t o t h e d e f i n i t i o n o f ( 3 . 2 1 ) .

If f(g)

2)

L b y s ( G ) , where k i s a n o n - n e g a t i v e i n t e g e r , U < r < p / ( p - 1 ) ,

f

1 < p c 2, U < s < 1, t h e n F o u r i e r s e r i e s o f f ( y ) s a t i s f i e s

p r o v i d e d k+s > ( ( 3 / p ) - 1 / 2 ) m + q ( r - ' + p - ' - l ) . I f f ( g ) f L b y s ( G ) , where k i s a n o n - n e g a t i v e i n t e g e r , 0 < r < 2,

3)

1< p <

Z,, U <

s < 1, t h e n F o u r i e r s e r i e s o f f ( y ) s a t i s f i e s

p r o v i d e d k+s > ( ( 3 / r ) + ( 3 / p ) - 3 ) m + q / p .

4)

I f f ( g ) 6 L b y s ( G ) , 1 < p < 2, 0 < r < p / ( p - l ) ,

0 < s < 1, k i s non-

negative integer, then

p r o v i d e d k+s > (3/p-3/2)m+q(l/(2r)+l/p-l). The p r i n c i p a l r e s u l t s abroad p a r a l l e l t o t h o s e on t h e c u b i c a l p a r t i a l sums of F o u r i e r s e r i e s i n t h i s s e c t i o n and t o t h o s e on t h e summations o f F o u r i e r s e r i e s i n s e c t i o n 2 a r e as f o l l o w s . In

[lo],

K.

Coifman and G. Weiss s t u d i e d t h e r e l a t i o n between t h e c e n t r a l

m u l t i p l i e r s f o r F o u r i e r s e r i e s on compact L i e g r o u p s and t h e m u l t i p l i e r s f o r multiple Fourier series. Let

They p r o v e d t h e f o l l o w i n g r e s u l t .

H be t h e C a r t a n s u b - a l g e b r a o f t h e L i e a l g e b r a o f a compact L i e g r o u p

exp be t h e e x p o n e n t i a l mapping, and

E

be t h e u n i t e l e m e n t o f G.

Ino ( d x m h ) C h ( r )

(3.22)

hEG

d e f i n e s t h e bounded m u l t i p l i e r on L ( H / e x p - l E ) , P

G,

If

then

1- mhdxXx(Y) x fG d e f i n e s t h e bounded c e n t r a l m u l t i p l i e r on L P ( G ) ,

(3.23) where p > 1.

I n addition, the

p r e c e d i n g c o n d i t i o n s a r e a l s o n e c e s s a r y f o r p = 1. I n (3.22),

'I E

H,

C,(T)

=

1 oew

eiB(a,ar),

(3.24)

S. Gong et al.

98

where W d e n o t e s t h e Weyl g r o u p , and B(

,

) represents t h e i n v a r i a n t inner

p r o d u c t on t h e L i e a l y e b r a f o r G. The d i f f e r e n c e o p e r a t o r

0

i n (3.22)

brinys

rn

where a l , a2

,..., a,

are a l l p o s i t i v e roots.

R. J . S t a n t o n and P. A. Tomas ( s e e 1121 and L131) d i s c u s s e d t h e p o l y h e d r a l p a r t i a l sums o f F o u r i e r s e r i e s on compact L i e y r o u p s d e f i n e d as f o l l o w s . Suppose t h a t 9 i s a c l o s e d connex p o l y h e d r o n w h i c h t a k e s t h e o r i g i n as i t s i n t e r i o r p o i n t and i s i n v a r i a n t under t h e t r a n s f o r m a t i o n o f t h e Weyl y r o u p i n t h e C a r t a n s u b a l y e b r a H o f t h e L i e a l y e b r a o f a compact L i e y r o u p G, and l e t Rt = { t x l x E R}.

They d e f i n e d t h e p o l y h e d r a l p a r t i a l sums o f f

6

L p ( G ) , p > 1,

by

and p r o v e d t h e f o l l o w i n y :

1) L e t G be a s i m p l y c o n n e c t e d s i m p l e compact L i e yroup, T be a maximal t o r u s of G, d i m G = n, dim T = q and Ly(G) be a l l c l a s s f u n c t i o n s i n L p ( G ) . p > 2 n / ( n + q ) and f

e

If

L ~ ( G ) ,t h e n S N f ( x ) a l m o s t e v e r y w h e r e c o n v e r g e s t o f .

2 ) When G, T, n, q a r e t h e same as I n l ) , and p < 2 n / ( n + q ) o r p > 2 n / ( n - q ) , t h e r e e x i s t s f E L y ( G ) such t h a t S N f ( x ) does n o t c o n v e r y e i n t h e sense o f L p norm.

3)

When G, T, n, q a r e a l s o t h e same as l ) , t h e r e e x i s t s a number p ( R ) ,

2 n / ( n t q ) < p(R)

(2n-2q+2)/(n-q+2)

such t h a t S N f ( x ) c o n v e r y e s i n t h e sense of

L p norm, t o f ( x ) f o r p ( R ) < p < p ( R ) ' and f

4)

6

Ly(ti).

When G i s a s i m p l e c o n n e c t e d s e m i - s i m p l e compact L i e yroup, t h e n t h e

r e s u l t s c o r r e s p o n d i n g t o 1)

-

3 ) can be composed by c o m b i n i n g t h e r e s u l t s o f

i t s s i m p l e subgroups. ) t h a t S N f ( x ) does n o t c o n v e r g e i n 5 ) When p # 2, t h e r e e x i s t s f E L ~ G such t h e sense o f L p norm.

6)

When p < 2, t h e r e e x i s t s f 6 Lp(G) such t h a t S N f ( x ) a l m o s t e v e r y w h e r e

does n o t c o n v e r g e t o f ( x ) .

R . A. Mayer ( s e e [18])

discussed F o u r i e r series f o r G = SU(2).

He p r o v e d

the followiny.

1) L e t f 6 C 1 ( t i ) , t h e n t h e F o u r i e r s e r i e s f o r f c o n v e r g e s u n i f o r m l y , and t h e r e e x i s t s g 6 C1(G) such t h a t i t s F o u r i e r s e r i e s does n o t c o n v e r g e absolutely. 2)

Let f

e L2(G) and f b e l o n g t o c l a s s C1 a l m o s t everywhere, t h e n t h e

F o u r i e r s e r i e s o f f a l m o s t e v e r y w h e r e c o n v e r g e s t o f.

Here f t h a t b e l o n g s t o

99

Harmonic Analysis on Classical Groups

c l a s s C1 a t one p o i n t means t h a t f i n a neighborhood o f t h e p o i n t i s equal t o a function i n c ~ ( G ) . 3)

L e t f E L1(G) and f be equal t o z e r o i n a neighborhood o f a p o i n t b e G.

Moreover, F o u r i e r s e r i e s Then

I;= p, nf(x)

f o r f s a t i s f i e s P n f ( b ) + 0, when n +

-.

p n f ( b ) converges t o zero.

L a t e r , Mayar s t u d i e d v a r i o u s problems about F o u r i e r s e r i e s on S U ( 2 ) systematically. I n [lY], (3.21))

PI.

E. T a y l o r discussed t h e a b s o l u t e converyence ( i n t h e sense o f

o f F o u r i e r s e r i e s on compact L i e groups and proved : l e t G be a compact

L i e yroup, dim G = n, and l e t s > n/4 be an i n t e y e r .

I f f e HZs and i n p a r t i -

c u l a r i f f E CZs(G), t h e n t h e F o u r i e r s e r i e s o f f converges a b s o l u t e l y and u n i f orml y

.

0. L. Ragozin (see 3 ) o f [20])

discussed t h e problem o f t h e a b s o l u t e con-

vergence o f F o u r i e r s e r i e s on compact L i e yroups i n t h e f o l l o w i n g sense and t h e problem o f t h e r e l a t i o n between t h e convergence and t h e d i f f e r e n t i a b i l i t y o f f :

where t h e meaning o f t h e r e l a t e d n o t a t i o n s i s t h e same as i n (3.19)

and (3.20),

and t r ( l C a I P ) i s d e f i n e d as f o l l o w s : L e t xl,

x2,

...,

be t h e c h a r a c t e r i s t i c r o o t s o f

non-negative and

Cayi.

Then t h e y a r e

d.

B. D r e s e l e r (see C161 and C171) s t u d i e d Lebesgue c o n s t a n t s f o r s p h e r i c a l p a r t i a l Sums o f F o u r i e r s e r i e s on compact L i e groups and proved t h a t t h e Lebesyue c o n s t a n t s a r e O(N(n-1)/2).

Moreover, he gave t h e e s t i m a t e s from above

and from below, n b e i n g t h e dimension o f t h e yroup.

4. Sumnation by S p h e r i c a l Means The d e f i n i t i o n o f summation by s p h e r i c a l means i n harmonic a n a l y s i s on u n i t a r y yroups and t h e r e l a t e d methods (see [6])

a r e w i d e l y used i n t h e

r e s e a r c h f o r harmonic a n a l y s i s on c l a s s i c a l groups and on compact L i e yroups. The s p h e r i c a l means summation o f F o u r i e r s e r i e s on u n i t a r y groups, essentially,

i s such a summation t h a t t h o s e terms o f F o u r i e r s e r i e s c o r r e s p o n d i n g t o

t h o s e f u n c t i o n s h a v i n g t h e same c h a r a c t e r i s t i c values o f L a p l a c e o p e r a t o r i n t h e r e p r e s e n t a t i v e r i n g o f a u n i t a r y group a r e m u l t i p l i e d by t h e same c o e f f i cient.

T h i s can e a s i l y be done by t a k i n g

adding a f a c t o r f u n c t i o n e x p ( - i ( n - l ) ( e l +

(4.4).

ak = fk + ( n - Z k + l ) / Z i n (4.1) and

...+en)/2)

t o the i n t e g r a l expression

S.Gong el al.

100

I n t h e r e s e a r c h work on t h e s p h e r i c a l means summation i n u n i t a r y groups, a method based on t h e F o u r i e r t r a n s f o r m a t i o n on C a r t a n s u b a l g e b r a s was e s t a blished.

As a C a r t a n s u b - a l g e b r a ,

u n d e r t h e i n v a r i a n t i n n e r p r o d u c t , con-

s t i t u t e s an E u c l i d e a n space, a v a r i e t y o f t o o l s o f t h e F o u r i e r t r a n s f o r m a t i o n i n t h e E u c l i d e a n space c a n be a p p l i e d .

Some r e s e a r c h e r s abroad such as

H.

S.

S t r i c h a r t z ( s e e C141) a d o p t e d a r e s e a r c h method, whose b a s i s i s t h e F o u r i e r t r a n s f o r m a t i o n on t h e L i e a l g e b r a .

C o m p a r a t i v e l y , t h e f o r m e r n o t o n l y can g i v e

an e x p l i c i t e x p r e s s i o n and r a t h e r a c c u r a t e r e s u l t s b u t a l s o can g i v e more F o r example, a w i d e c l a s s o f bounded o p e r a t o r s

r e s u l t s t o a l o t o f problems. on L ( G ) i n Theorem 4.12

( 1 ) w h i c h i s e s t a b l i s h e d by t h e methods on u n i t a r y

y r o u p s c a n n o t be o b t a i n e d by t h e methods on L i e a l g e b r a s i n [14].

But t h e

l a t t e r c e r t a i n l y has some advantages o v e r t h e f o r m e r i n some r e s p e c t s . F o r F o u r i e r s e r i e s ( l . Y ) y we c o n s i d e r t h e

L e t u ( U ) be i n t e g r a b l e on Un. f o l l o w i n g sum

I:

I:

m

fl>.,.>f

e;+.. ek = f k +n-k, k = l,2,...yn. L e t 4(t) be a f u n c t i o n on 0 <

N(f)tr(CfAf(U)) n

.+aE=m,

where

g i v e s us t h e means o f

4(6/R

O b v i o u s l y , when u(U

4.1)

-1

< -, c o n t i n u o u s a t t = 0 and a ( 0 ) = 1..

t

d e f i n e d as f o l l o w s :

1 4(JG/R)

m

I:

fl>" . > f

e 2l + . ..+ez=m

i s i n t e g r a b l e and R i s a c o n s t a n t , (4.2)

c o n v e r g e n t f o r a l m o s t a l l U E Un,

f

If t h e l i m i t o f ( 4 . 2 ) e x i s t s f o r R + I n (4.2),

i s uniformly

provided

1 (@(J;/R)N(f)( 4-summable t o a l i m i t .

N(f)tr(CfAf(U)). n

m

<

+

(4.3)

m.

then F o u r i e r s e r i e s f o r u(U) i s c a l l e d

b = n(n-1)(2n-l

/6.

Taking

where J s ( u ) i s t h e Bessel f u n c t i o n o f o r d e r s o f t h e f i r s t k i n d .

THEOREM 4.1. e x p r e s s e d as

(see

C61).

I f u ( U ) i s i n t e g r a b l e on Un,

t h e n (4.2)

can be

Harmonic Analysis on Classical Groups

a

D(

x

101

a

acl

(4.4)

acn

I , . . . , -

Here 6 ( t ) must s a t i s f y t h e f o l l o w i n g c o n d i t i o n s : 1)

b ( t ) i s a b s o l u t e l y c o n t i n u o u s on any d e f i n i t e i n t e r v a l ,

2)

jmI $ ( t ) I t ( n - 1 ) i 2 d t

where 0 < j, (4.4)

<

0

,...,j n <

(4.5)

-9

... + j n =

jl +

n-1,

n(n-1)/2.

can be r e w r i t t e n as

=

xf

(A)CfN( f)-l,

where A i s a d i a g o n a l m a t r i x and i t s d i a g o n a l e l e m e n t s a r e e

i5 1

,...,e i s n .

From t h i s , we o b t a i n

1

J 'n

s ~ ( ~ ; u ) A ~ ( u =~ )c o + ( ~ ~ / R ) ~ ( J w R ) - ' . f

Thus t h e F o u r i e r s e r i e s of (4.4) all

R > 0 is Si(u;U) By (4.3),

6(fi/R)-l

1 6(~/R)N(f)tr(CfAf(U)).

8

Un,

t h u s (4.2)

a )(Hb a ,..., -

acn

and (4.4)

(n-2)/2

(4.8)

1

t h e s e r i e s on t h e r i g h t s i d e o f (4.8)

almost every U As D(

-

w h i c h a r e i n t e g r a b l e f u n c t i o n s on Un f o r

1 1 1 l-n)

( 15 )

i s a b s o l u t e l y convergent f o r

a r e equal f o r almost, every u

6

Un.

can be c a l c u l a t e d by r e c u r r e n c e

formula i t takes i t s o r i g i n a l s i g n o r t h e o p p o s i t e s i g n under t h e permutation (cl

,...,5,)

+

(cj

1

,...,5 .

'n

) a c c o r d i n g t o t h e p e r m u t a t i o n b e i n g even o r odd.

102

S. Gong et al.

Thus i t i s e q u a l t o (-1)n(n-1)/2D(cl,...,cn)H and (4.7)

6 (n

a r e equal.

F o r 6 ( t ) i n t h e s p h e r i c a l means ( 4 . 2 ) ,

-2)/2

2 ( ~ E ( ) I ~ ,I i ~. e .- ~(4.4)

t h e most i n t e r e s t i n y examples a r e t h e

f o l 1owi ny :

1)

6 ( t ) = e-t,

2)

g ( t ) = e-t2,

3)

6(t) =

{

t h e Poisson-Abel summation, t h e Gauss-Sommerfeld summation,

:1-t2)6

Then, i n t h e A b e l - ,

for

o <

for

1 < t,

< 1,

t

t h e Gauss-,

t h e R i e s z summation o f o r d e r 6.

and t h e Riesz-summation o f o r d e r 6 o f

F o u r i e r s e r i e s f o r u ( U ) we c o n s i d e r (4.9) el+. 2 G s~(u;u) =

1 e- in/ R

tb

/R

m

..+eE=m

1 ... 'f,

fl'

N(f)tr(CfAf(U)),

( 4.10 )

.

el+. 2 .+e:=m and

Si(u;U)

=

fl'".'fn m =

respectively.

2 el+

1

( 1 - b / R 2 ) - 6 ( 1-m/R2)& N( f ) t r ( C f A f ( U ) )

(4.11)

2 2 ...fen
I t i s o b v i o u s t h a t t h e s e t h r e e summations s a t i s f y t h e c o n d i t i o n s

i n Theorem 4.1.

THEOREM 4.2.

(see [S]).

means S i ( u , U )

o f t h e F o u r i e r s e r i e s f o r u(U) converges t o u ( U ) u n i f o r m l y .

2)

L e t u ( U ) E Lp(Un).

for R +

3)

m

1)

L e t u ( U ) be c o n t i n u o u s on Un.

Then SR(u;U) A 6 Lp(Un) and S i ( u ; U )

i n t h e norm o f Lp(Un), where p > 1; and I l S i ( u ; U ) l l

L e t u ( U ) be i n t e g r a b l e .

Then S;(u;U)

L e t u(U) E L i p a .

Then

A (SR(u;U) i f 0 < a < 1,and

ISR(u;U) A i f a = 1.

-

-

P

converges t o u ( U )

u ( U ) I < AIK-'

u ( U ) ( < A2R-'log

R,

-

< AoIlu(U)IIp.

converges t o u(U) f o r R +

everywhere.

4)

Then t h e Abel

almost

Huniionic Anu1,vsis O I I Classical Groups (see [S]).

THEOREM 4.3. Si(u;U)

1 ) I f u ( U ) i s c o n t i n u o u s on U,

103 t h e n t h e Gauss means

o f i t s F o u r i e r s e r i e s u n i f o r m l y converyes t o u(U) f o r R +

modulus o f c o n t i n u i t y o f u ( U ) i s w ( t ) ,

-

IS;(u;U)

2) I f u(U)

B

-,

and i f t h e

then

u ( U ) ( < A3w(K

-1

),

G t h e n SR(u;U) 6 Lp(Un) and

Lp(U,),

G IIS~(U;U)II < A 4 ~ l u ( U ) i ~ P P’ and SG(u;U) c o n v e r y e s t o u ( U ) f o r K + i n t h e norm o f Lp(U,), where p > 1. K ti 3 ) I f u ( U ) i s i n t e g r a b l e on Un, t h e n SR(u;U) c o n v e r g e s t o u ( U ) f o r R +

-

a l m o s t everywhere.

THEOREM 4.4.

then

converges t o u(u) u n i f o r m l y , f o r R +

s;(u;u)

1)

I f 6 > (n2-1)/2,

(see [S]).

And i f u ( U ) 6 L i p a, (1 < a < 1, t h e n

U.,

2)

+

if

U(U)

a)

(SR(u;U)

-

u ( U ) I < AgK - 6 + ( n 2 - 1 ) / 2 ,

b)

lSi(u;U)

-

u ( U ) ) < A6R-alog R , i f a + ( n 2 - 1 ) / 2 = 6;

c)

IS;(~;U)

-

u ( u ) ( < A ~ R - ~ i, f a

I f u(U)

Lp(Un),

3)

6

-

6 Lp(U,),

for R +

-,

p > 1, t h e n S;(U;u)

i s continuous on

( n2 - 1 ) / 2 > 6 ;

(n2-1)/2 < 6 . c o n v e r g e s t o u ( U ) i n t h e norm o f

and I l S ~ ( u ; U ) i lP < A811u(U)II P’

I f u(U) i s i n t e g r a b l e on Un,

everywhere f o r R +

+

if a

-

-.

I n Theorems 4.2, 4.3

and 4.4,

then Si(u;U)

t h e numbers Ao,

c o n v e r g e s t o u(U

A1

a1 inos t

ndependent o f

R.

THEOREM 4.5.

(see [6])

V a l , and (4.3),

i f U < (51 < 1/R,

where p > 0, Then,

(4.5),

L e t a ( t ) be a b s o l u t e l y c o n t nuous on any f i n i t e i n t e r and (4.6)

Eloreover, we have

and

i f 1/R < (51 <

for R +

be v a l i d f o r a ( t ) .

-,

Si(u;U)

m.

u n i f o r m l y converges t o u(U) p r o v i d e d u(U) i s

continuous. As i n t h e c a s e o f r o t a t i o n y r o u p s , Wang S h i k u n and Dong Daozheng d i s c u s s e d t h e summation o f F o u r i e r s e r i e s o n r o t a t i o n g r o u p s by s p h e r i c a l means. proved:

They

S. Gong e l a1

104

THEOREM 4.6.

u(r)

(Wang S h i k u n and Dong Daozheng see [l]). Let

on SO(n) and

1

m

where b = 1' + 2'

+

le(JTiIR)N(m)

... + ( k - 1 ) '

I

<

+

be i n t e g r a b l e

-,

(4.12)

i f n = 2k, and b = ( 1 / 2 ) 2 +

... + ( k - 1 / 2 ) 2

if

n = 2 k + l , and t h e n t h e i n t e g r a l r e p r e s e n t a t i o n o f t h e s p h e r i c a l means o f Fourier series for

u(r)

is

(4.14)

on any f i n i t e i n t e r v a l ;

= 0 , where 0

THEOREM 4.7.

(Wang S h i k u n and Dong Daozheng, see [ l ] ) .

< j,

,...,j k <

n-2.

By t a k i n g t h e above-

m e n t i o n e d t h r e e f u n c t i o n s as + ( t ) , d e f i n e t h e s o - c a l l e d t h e Abel, t h e Gauss and t h e R i e s z summations o f o r d e r 6 o f F o u r i e r s e r i e s on r o t a t i o n y r o u p s r e s p e c tively.

F o r t h e s e t h r e e summations t h e f o l l o w i n g r e s u l t s a r e v a l i d where

6 > n(n-1)/4-1/2

1)

u(r)

2)

These t h r e e summations u n i f o r m l y c o n v e r g e t o

u(r)

u(r)

for R +

m,

provided

i s c o n t i n u o u s on s O ( n ) .

Si(u;r),

provided 3)

i s needed:

u(r)

Si(U;r),

S i ( u ; r ) , i n t h e of L P ( S O ( n ) ) , p > 1, c o n v e r g e t o

u(r),

E LP(SO(n)).

G SAR ( u ; r ) , SR(u;T),

Si(u;r) a l m o s t e v e r y w h e r e c o n v e r g e t o u ( r ) , p r o v i d e d

i s integrable.

THEOREM 4.8. on SO(n),

(Wang S h i k u n and Dong Daozheng, see [ l ] ) .

Let

u(r)

be c o n t i n u o u s

and $ ( t ) be a b s o l u t e l y c o n t i n u o u s i n any f i n i t e i n t e r v a l and s a t i s f y

Harmonic Analysis on Classical Groups

4)

ti,

a ( --

a

a%

i f 151 > 1 / R ,

,..., -

Ht/2-1(1c1R) = O(R-p-llcl-p-kn+k 2 );

--k-l

Id

ack

t h e n S:(u;r)

105

u n i f o r m l y converges t o

u(r).

He Zuqi and Chen Guangxiao d i s c u s s e d t h e summation by s p h e r i c a l means on u n i t a r y s y m p l e c t i c g r o u p s and o b t a i n e d t h e f o l l o w i n y r e s u l t s .

THEOREM 4.9.

(He Zuqi and Cheny Guangxiao, see E l ] ) .

L e t u U) be i n t e g r a b l e on

USP(2n) and

2)

+ ( t ) i s a b s o l u t e l y c o n t i n u o u s i n any f i n i t e i n t e r v a l

Then t h e s p h e r i c a l means o f F o u r i e r s e r i e s f o r u ( U ) (4.15) al+. 2 whose i n t e g r a l r e p r e s e n t a t i o n i s Si(u;U)

=

where Q ( x l

( - i I n R ( 2 n ) - n / 2 2-n n!D(nL,...,lL)n!+(JR)

,...,xn)

= xlx 2 . . . ~ n D ( ~ :

J

..+e:=m

-... J

-m

m

$(,e

is1

,...,e

is,

)

-m

,...,x n2 ) ,

tk = fk+n+l-k,

k = 1,2

,...,n,

and

b = n(n+1)(2n+1)/6. By t a k i n g t h e aboveA G m e n t i o n e d t h r e e f u n c t i o n s as + ( t ) , s p h e r i c a l means SR(u;U), SR(u;U) and

THEOREM 4.10. Si(u;U)

(He Zuqi and Chen Guangxiao,

see [ l ] ) .

o f F o u r i e r s e r i e s f o r u(U) a r e d e f i n e d r e s p e c t i v e l y , and t h e f o l l o w i n g

r e s u l t s a r e v a l i d ( f o r t h e R i e s z means, t h e c o n d i t i o n 6 > n2 + ( n - 1 ) / 2 needed) :

is

S. Gong et al.

106

1 cont

The t h r e e s p h e r i c a l means c o n v e r g e t o u(U) f o r R + flUOUS

m

i f u(U) i s

on USP(2n).

F o r p > 1, t h e t h r e e s p h e r i c a l means c o n v e r g e t o u ( U ) i n t h e norm o f

2

L ~ ( u s P ( ~ i~ f) )U ( U ) e L ~ ( u s P ( ~ ~ ) ) . 3)

The t h r e e s p h e r i c a l means a l m o s t e v e r y w h e r e c o n v e r g e t o u(U) f o r R +

m

i f u(U) i s i n t e y r a b l e .

THEOREM 4.11. (He Zuqi and Chen Guanxiao, see [l]). c o n d i t i o n s i n Theorem 4.9.

where p > 0 and u(U) i s c o n t i n u o u s . for R +

Let $ ( t ) satisfy the

Moreover

Then S$(u;U)

u n i f o r m l y converges t o u(U)

m.

L i S h i x i o n g and Zheng Xucan d i s c u s s e d t h e s p h e r i c a l means and t h e more g e n e r a l means o f F o u r i e r s e r i e s on compact L i e groups. L e t G be a compact L i e group o f d i m e n s i o n n, T be a maximal t o r u s o f dimens i o n q o f G, rn = ( n - q ) / 2 , B(

,

H be t h e C a r t a n s u b - a l g e b r a o f t h e L i e a l g e b r a o f G ,

) be t h e i n v a r i a n t i n n e r p r o d u c t on t h e L i e a l g e b r a o f G , and (

,

) * be

t h e special i n v a r i a n t i n n e r product which i s c a l l e d quasi K i l l i n g - C a r t a n form on compact L i e a l g e b r a s .

The r e l a t e d d e f i n i t i o n can be found i n " F o u r i e r

a n a l y s i s on compact L i e g r o u p s " t o appear i n "Advances i n Flathematics ( i n C h i n e s e ) [21]. Let f(g)

6

L(G).

We c o n s i d e r t h e f o l l o w i n g means o f F o u r i e r s e r i e s f o r f ( g ) .

1) L e t $ ( t ) , H!(t)

and W;(t)

d e f i n e as b e f o r e and c o n s i d e r t h e s p h e r i c a l

means o f F o u r i e r s e r i e s ( 3 . 2 0 ) f o r f ( g )

(4.17-1) t ;(h) 6

6

L ( H ) , ;(h)

be i n v a r i a n t under t h e t r a n s f o r m a t i o n o f t h e Weyl

H,

$ ( h ) = ( 2 ~ ) - ~ / ./' $(y)e-iB(hyY)dy H d e r t h e means

lA o(

AEG

I$( ft. ) - l d X f * X X ( S ) ,

(4.17-2) (4.17-3)

(4.17-4)

I07

Harmonic AnaI.vsis on Classical Groups Take $ ( h ) = W$2-l(lhl) (4.17-1),

i n (4.17-2).

where

( A + B I = lB(A+B,

I t i s o b v i o u s t h a t (4.17-2)

becomes

A+B)) 1/2 ,

and, as a f u n c t i o n on H, g ( h ) i s i n v a r i a n t u n d e r t h e t r a n s f o r m a t i o n o f t h e Weyl yroup.

Thus g ( h ) can be u n i q u e l y e x t e n d e d as a f u n c t i o n on t h e L i e a l y e b r a f o r

G, t h e v a l u e s o f w h i c h a r e @ ( a d h ) = g ( h ) f o r h E H and y Y L e t a1

,...,am be

a l l p o s i t i v e r o o t s on H; p ( h ) = n j = l

be t h e o r t h o n o r m a l b a s i s f o r B( and h = xlXl+x2X2+

...+ x X

q q

,

) i n H;

a = ax

Y

Q(h) = ?r

ZoeW

(4.17-4)

and C,

E

e

G.

B(h,aj);

a + x2

1 ax1

a +,,,+ ax2

Y

(exp h) =

I f(gt

G

G, exp be t h e e x p o n e n t i a l mapping; A ( h ) = n y = 1 ( 2 i iB(ho,oh)

,

ho

X1,X2

,...,X 9 x a q axq’

be a p o i n t i n H; a g a i n l e t W d e n o t e t h e Weyl y r o u p ,

)111 be t h e o r d e r o f t h e Weyl y r o u p ; J, ( h ) = h E H, y , t

x

E

e H.

exp(h)t-’)dt, sin

B(h,ai));

Then t h e i n t e y r a l e x p r e s s i o n s f o r (4.17-1)

are respectively

depends o n l y on t h e L i e a l g e b r a .

For t h e means (4.17-1) e x p r e s s i o n s S!,R(f;g)

THEOREH 4.12.

, ,+

%

(4.17-4)

Si,R(f;g),

and f o r t h e i r c o r r e s p o n d i n g i n t e g r a l the following results are valid.

( L i S h i x i o n g and Zheng Xuean).

v a t i v e s o f up t o m - t i m e s on H.

L e t $ ( h ) h a v e L1 p a r t i a l d e r i -

Then t h e f o l l o w i n g h o l d .

S. Gong et al.

108

1 ) I f f ( g ) E L p ( G ) , p > 1, f o r j = 1,2,3, o p e r a t i o n s on Lp(G) and

t h e n SQ,,(f;g)

a r e bounded l i n e a r

l l S ~ , R ( f ; y ) i i p < A(G,$,j ,R) ilfll P ' and S'? ( f ; g ) i s r e y a r d e d as an i n t e g r a b l e f u n c t i o n f o r w h i c h t h e F o u r i e r J ,R s e r i e s i s j u s t (4.17-j).

&

1, and i f P ( h ) P ( ) { $ ( h ) } 6 L(H), then f o r I f f ( g ) E Lp(G), p S'? ( f ; g ) i s a bounded l i n e a r o p e r a t o r on Lp(G) and J ,R

2)

j = 1,2,3,4,

and S'? ( f ; g ) i s r e g a r d e d as an i n t e g r a b l e f u n c t i o n f o r w h i c h t h e F o u r i e r J ,R s e r i e s i s j u s t (4.17-j).

&

) { $ ( h ) } ( < A ( l + (hI)-"', and j = 1,2,3,4, 3) I f IP(h)-lP( b e s i d e s 2 ) o f t h i s theorem, t h e f o l l o w i n g r e s u l t s a r e v a l i d . a)

SQ,,(f;y)

almost everywhere converges t o f ( g )

b)

SQ,R(f;g)

u n i f o r m l y converges t o f ( g ) f o r R +

C)

SUP

R>U

M f ( y ) = sup r>O

(sj,R(f;g)I

/

B(g;r)

< A(G,$,j)(Mf(Y)

If(t)(dt(B(g;r)(-',

+

/

G

h(t )

6

-

-'

E

L(G) f o r R +

> 0, t h e n

m.

i f f ( g ) i s continuous.

I I f ( g t - ' ) 1 d t ) , where

and B ( g r ) d e n o t e s a l l t

E

G from which

t h e Riemann d i s t a n c e t o g i s l e s s t h a n r;

[ { s uR p I S ? ,R ( f ; g ) )

d) 4)

If

1-

A€G

I$((A+e)/R)ldA <

1

A€G

(4.17-j)

> y } l < A(G,$,j)y-lllfllLl.

+

m,

then,

i n t h e sense o f t h a t

l$((A+E)/R)dAf*xX(Y)

1

<

+

m,

a b s o l u t e l y c o n v e r y e s f o r a l m o s t e v e r y g 6 G and j = 1,Z.

And, i n t h e

i s e q u a l t o S'? ( f ; g ) f o r a l m o s t e v e r y y 6 G and j = 1,2, J sR G(h) s a t i s f i e s t h e f i r s t c o n d i t i o n , where f ( g ) 6 L(G) b e s i d e s t h e above

meantime, ( 4 . 1 7 - j )

if

mentioned c o n d i t i o n s .

5)

(4.17-j)

j = 1,2,3,4,

i n t h e sense o f 4 ) a b s o l u t e l y c o n v e r y e s f o r a l l R and

i f ( $ ( h ) I < C ( l t (h()-n'z-q/2-E,

E

> 0.

And, i n t h e meantime,

i s equal t o S'? ( f ; g ) f o r a l m o s t e v e r y g 6 G and j = 1,2,3 o r j = 4, J ,R i f $ ( h ) s a t i s f i e s t h e f i r s t c o n d i t i o n o r t h e c o n d i t i o n i n 3) r e s p e c t i v e l y , (4.17-j)

b e s i d e s t h e above m e n t i o n e d c o n d i t i o n s .

6 ) From t h e P o i s s o n summation f o r m u l a t h e summation k e r n e l s K ? ( 9 ) c a n be J YR deduced w h i c h s a t i s f i e s

Harmonic Analysis on Classical Groups

S$

THEOREM 4.13. U

1

f ( g t - l ) KQ,R(t)dt. G Take + ( t ) = ( 1 - t 2 k ) 6 , ( L i S h i x i o n g and Zheng Xuean).

J ,R

(f;g) =

109

t < 1 and 0 f o r t > 1, k b e i n g a p o s i t i v e i n t e g e r .

Thus (4.17-1)

(4.18) for defines

t h e R i e s z summation of o r d e r 6 and d e g r e e 2k o f F o u r i e r s e r i e s on compact L i e When k = 1, i t i s t h e u s u a l 1 R i e s z S Z k s 6 ( f ; g ) denotes S2k*6(f;g). 1,R summation d e n o t e d by S i ( f ; g ) . Then S i k S 6 ( f ; g ) s a t i s f i e s t h e f o l l o w i n y :

groups.

i s valid f o r Sik9&(f;g) i f 6 > (n-l)/2.

1)

The c o n c l u s i o n of Theorem 4.12

2)

I f f ( g ) i s c o n t i n u o u s on G and 6 > ( n - 1 ) / 2 t h e n 1Sik"(f;g)

3)

The s a t u r a t i o n o r d e r o f

THEOREM 4.14.

-

f(g)

Siky6

1

< A(G,k,G)w(f;l/R).

i s R-2k.

( L i S h i x i o n g and Zheng Xuean).

I f 6 > (n-1)/2,

then

s < 2k-1.

< A(G,k,G)

1

x+Bl
i f f ( g ) c C2k(G).

Ilfl12kR-2k

dxtr(C,A,(g))

-

f ( g ) l l m } , C x an a r b i t r a r y

and i n d e p e n d e n t o f f ( g ) .

From 1 ) and 2 ) o f t h i s

theorem, t h e J a c k s o n Theorem f o l l o w s : ER(f) i f f(g)

6

CksW(G).

<

A(G,k,p)

liflk,wR-kw(l/R),

B e s i d e s , t h e B e r n s t e i n Theorem c a n be d i r e c t l y deduced b y

t h e u n i t a r y r e p r e s e n t a t i o n s f o r compact L i e groups. 4)

L e t 11 > ( n - 1 ) / 2 ,

M ' where T k ( y ) = f r o m 1,2,..

k

1

x. y

M be an i n t e g e r , and

and any o f xl,

x2,

..., x

j =1 .,M. Then

M IIVR(f;Y)

P4 where sup llVRU < + R>O

THEOREM 4.15.

i s a sum o f k numbers c h o s e n $4

-.

-

f(y)( <

PI

( S U P IIVRII +

R>U

( L i S h i x i o n g , Fan Dashan and Zheng Xuean).

Then t h e k e r n e l s o f R i e s z means o f ( 2 k , 6 ) s a t i s f y 1)

2k,6, nKR

l)ER(f),

(g)iill

A(G,k)

l o y R,

( ( s e e (4.18)).

Take 6, = ( n - 1 ) / 2 .

S. Gong el al.

110

6

nsup ) S i k ' 6 ( f , y ) 1 i i p R>O

b)

S i k s 6 ( f ; y ) c o n v e r g e s t o f ( y ) a l m o s t everywhere;

c)

l i m iisiky6(f;g) I?+-

-

)iitilp;

= 0.

f(g)ii,

I

f ( y ) d y , where g,y a G. B(g;r) G t h e r e e x i s t s r o > 0 such t h a t

4) g

< A(G,P,k

a)

Let f*(g;r)

=

f*(g;r+2s)

-

+ f*(g;r)

2f*(g;r+s)

I f f o r almost every

= o(s/log

s)

i s v a l i d u n i f o r m l y f o r s < r < ro, t h e n t h e F o l l o w i n g r e s u l t o f t h e Salem t y p e 2 k ,6 i s v a l i d : sR O(f;g) converges t o f ( g ) ( f o r R + W ) almost everywhere i f I t ) l o g + l f ( i s i n t e g r a l f o r G b e i n g a t o r u s of d i m e n s i o n i n t e g r a l f o r G b e i n g o t h e r compact L i e group, Dini-,

n >

2 or i f f i s

S i m i l a r l y , we can g i v e t h e

t h e J o r d a n - and t h e L e b e s g u e - t e s t f o r S ~ " ' " O ( f ; g )

on compact L i e g r o u p s

by use o f t h e f u n c t i o n f * ( y ; r ) .

E. PI. S t e i n ( s e e [ 9 ] ) d i s c u s s e d t h e f o l l o w i n y s p h e r i c a l means o f F o u r i e r

where x E G, f E L ( G ) , and he p r o v e d

2)

where t > 0, f ( x ) € Lp(G), p > 1. P' p t i s a s e l f - c o n j u y a t e o p e r a t o r on L ~ ( G ) .

3)

f > 0 i m p l i e s t h a t ptf

4)

l i m -P= t f - f

llPtfUp c Ufll

1)

t+O

t

-(-A) l / Z f

0.

,

where

5) u(t;x) equation

I Ptf(x)

E C"(Gx(0,-)),

( 6) XI,

u(t;x)

X2,

and a l s o u ( t ; x )

5 2

+

A)U

I 0.

converges t o f ( x ) f o r t + 0 i n t h e norm o f L ( G ) , where

..., Xn

i s a b a s i s o f t h e L i e a l g e b r a o f G,

n A =

1

i,J=l

s a t i s f i e s t h e Laplace

( a i j ) = (-B(Xi,

n

a..X.x., 'J 1 J

AA ( x ) = -p A ( x ) , A f ( x ) = A X

1

a. . X . x . f . 1J 1 J

i ,j=1

L e t f be a r e a l v a l u e d f u n c t i o n w h i c h b e l o n g s t o C"(G)

and d e f i n e

Xj))".

Harrnonic Analysis on Classical Groups

( v f (2 ( x ) If f

6

Cm(tix(O,-)),

111

n a..(xif)(x.f). i ,j=1 1J J

1

=

then

S t e i n d e f i n e d t h e L i t t l e w o o d - P a l e y f u n c t i o n o f f f Lp(G) as

Then E. M.

m

I0 t l v u ( t ; x ) ( 2 d t ) 1 / 2

(

Y(f)(X) =

9

and p r o v e d t h e f o l l o w i n g :

7 ) Let f 6 Lp(G), 1 < P < Ap such t h a t

-.

Then g ( f )

IIg(f)ll Conversely, i f

I f(x)dx

= 0,

G

< A Ilfll P

Lp(G) and t h e r e e x i s t s a c o n s t a n t

P '

then there e x i s t s a constant B Ilfllp

8)

P

6

P

such t h a t

< Bpllg(f)llp.

L e t t h e R i e s z t r a n s f o r m a t i o n on G be j =1,2

K . f = X.(-A)-"'f, J J

...,n,

Then R . , j = 1,2, J from which f o l l o w s

where f E C"(G).

1 < p < -,

J. L. C l e r c ( s e e [ l l ] )

,...,n,

a r e bounded o p e r a t o r s on Lp(G) f o r

d i s c u s s e d t h e summation o f F o u r i e r s e r i e s on compact

L i e y r o u p s by R i e s z means o f o r d e r 6.

H i s m a i n r e s u l t s a r e as f o l l o w s :

L e t G be a compact L i e g r o u p o f d i m e n s i o n n and r a n k q, D ( e x p h ) be C l e y l ' s f u n c t i o n o f G and then, 1) 2)

3)

S i f + f f o r 6 > ( n - 1 ) / 2 i n t h e norm o f L p ( G ) , p sup ( S i f ( x ) I < C ( M f ( x )

f

K*lf((x)),

R>O

1.

6 > (n-l)/2.

If 6 > ( n - 1 ) / 2 , f E L(G) and m i s t h e Haar measure, t h e n m{sup I S i f I > a] < A llflll , R

and, from t h i s , S i f c o n v e r y e s t o f a l m o s t e v e r y w h e r e ;

4) that

I f 1 < p < 2, 6 > ( n - l ) ( l / p - 1 / 2 ) ,

IlSUP

R

R.

S.

t h e n t h e r e e x i s t s a c o n s t a n t Ap such

6

I S R f ( II

P

< A

P

llfll

P

.

S t r i c h a r t z ( s e e C141) d i s c u s s e d t h e m u l t i p l i e r t r a n s f o r m a t i o n on

compact L i e a l g e b r a s and groups.

S. Gong et al.

112

L e t G be a compact L i e y r o u p and $I be i t s L i e a l g e b r a , H be a C a r t a n subalgebra o f

9,

dp

be ad- n v a r i a n t f i n i t e measure o n

9.

E s p e c i a l l y , when dp

i s absolutely continuous, t h e r e e x i s t s a f u n c t i o n F(x), x E i n t e g r a b l e and a d - i n v a r i a t (i.e.

9,

which i s

F ( h ) ( P ( h ) I 2 i s i n t e g r a b l e on ti, h E H such

t h a t dp = F ( x ) d x . R. S. S t r i c h a r t z p r o v e d :

1)

If

then

$(A)

or

=

$(A) =

(*I J @ ( A + B - ~ ~ ~ B(**I) ~ Y @(A+B)

G

a r e bounded o p e r a t o r s on L ( G ) . 2 ) L e t @ ( x ) be t h e same as i n 1 ) and d e f i n e o r ( x ) = @ ( x / r ) . an o p e r a t o r 0P(@) on :

and ( * ) o r ( * * ) d e f i n e s a n o p e r a t o r o p ( $ ) on G. t h a t 0 P ( @ ) i s bounded on L ( D.

L. R a g o z i n ( s e e [ n o ] ) ,

9)is

Then d e f i n e s

Then t h e n e c e s s a r y c o n d i t i o n

t h a t o p ( g r ) i s u n i f o r m l y bounded when r +

m.

u s i n g i m b e d d i n g method i n t o t h e E u c l i d e a n space,

p r o v e d t h e Jackson Theorem, t h e B e r n s t e i n Theorem and o t h e r r e s u l t s on compact L i e groups and on compact homoyeneous spaces. As t o t h e harmonic a n a l y s i s on u n i t a r y groups and i t s e x t e n s i o n on c l a s s i c a l y r o u p s and compact L i e groups, t h e r e a r e many r e s u l t s such as : a v a r i e t y o f theorems o f Tauber t y p e , a v a r i e t y o f p r o b l e m s on how t o s t u d y t h e h a r m o n i c a n a l y s i s on c l a s s i c a l domains t h r o u y h t h e harmonic a n a l y s i s on c l a s s i c a l yroups, and many r e s u l t s on t h e a p p r o x i m a t i o n t h e o r y . omitted,

A l l these r e s u l t s are

f o r w h i c h t h e r e a d e r s a r e r e f e r r e d t o [l] - [ 6 ] and o t h e r a r t i c l e s .

REFERENCES

El1

Gony Sheny (Kuny Sun), Harmonic A n a l y s i s on C l a s s i c a l Gr-0ups ( i n Chinese S c i e n c e Press, B e i j i n g China, 1983. , Acta. Math. S i n i c a , 1 0 ( 1 Y 6 0 ) , 239-261 ( i n Chi nese c21 , i b i d 1 2 ( 1 9 6 2 ) , 17-31 ( i n C h i n e s e ) , C31 , i b i d 1 3 ( 1 9 6 3 ) , 152-161 ( i n C h i n e s e ) . C41 , i b i d 1 3 ( 1 9 6 3 ) , 323-331 ( i n C h i n e s e ) . [51 , i b i d 15(1Y65), 305-325 ( i n C h i n e s e ) . C61 1 7 1 2 h o n g J i a q i n g , J o u r n a l o f Chinese U n i v e r s i t y o f S c i e n c e and ’echnol ogy , 9(197Y), 31-43. [ 8 ] Gony Sheny, J o u r n a l o f Chinese U n i v e r s i t y o f S c i e n c e and Technology, 9 ( 1 9 7 9 ) , 25-30. [9] S t e i n , E. M., Annals i n Math. Study, P r i n c e t o n , 1970, No. 63. [ l o ] Coifman, R . & Weiss. G., B u l l . Amer. Math. SOC. 8 0 ( 1 9 7 4 ) , 124-126. [ll] C l e r c , J. L., Ann. I n s t . F o u r i e r . Grenoble, 2 4 ( 1 9 7 4 ) , 1:14Y-172. [12] S t a n t o n , R. J., Trans. Amer. Math. SOC. 218(1976), 61-81.

Harmonic Analysis on Classical Groups

[13] [14] [l5] [16]

[I71

[18]

[lY] [20] c211

c221 [23]

S t a n t o n , R. J . & Tomas, P. A., Amer. J. Math. 1 0 0 ( 1 9 7 8 ) , 477-493. S t r i c h a r t z , R. S., T r a n s . Amer. l l a t h . SOC. 1 9 3 ( 1 9 7 4 ) , 99-110. Weiss, N. J . , Amer. J. Math. 9 4 ( 1 9 7 2 ) , 1U3-118. D r e s e l e r , R., M a n u s c r i p t a Math. 3 1 ( 1 Y 8 0 ) , 17-23. , F o u r i e r A n a l y s i s and A p p r o x i m a t i o n Theory, Ed. G. A l e x i t s and P. Turan, V o l . I ( 1 9 7 6 ) , 327-342. Mayer, R. A., Duke Math. J . 3 4 ( 1 Y 6 7 ) , 549-554. T a l o r , M. E., Amer. Math. SOC. 1 Y ( 1 9 6 8 ) , 1103-1105. K a y o z i n , D. L., Trans. Amer. Math. SOC. 1 5 0 ( 1 9 7 0 ) , 41-53. , I l a t h . Ann. 1 9 5 ( 1 9 7 2 ) , 87-94. , i b i d , 2 1 9 ( 1 9 7 6 ) , 1-11. Zheny Xue An, Advances i n Math., V o l . 1 3 , 2 ( 1 9 8 4 ) , 103-118.

113