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Estimates for Finite Expansions of Gegenbauer and Jacobi Polynomials
Recent Progress in Fourier Analysis I. Peral and J.-L.Rubio de Francia (Editors) 0 Elsevier Sdence Publishers B.V. (North-Holland), 1985
ESTIMATES FOR FINITE EXPANSIONS OF GEGENBAUER AND JACOB1 POLYNOMIALS Mitchell H. Taibleson* Washington University in St. Louis St. Louis, Missouri 63130
Estimates will be given for certain Gegenbauer polynomial and Jacobi polynomial expansions. The primary interest is on expansions of a form that are needed to extend a result of E.M. Stein [6]. Stein shows that there is a function h E H 1 (Rn) (resp., h E H 1 (Tn1) such that the Bochner-Riesz means of h at the critical index (n-1)/2 diverge a.e. He accomplishes this as a consequence of the construction of a sequence {hN) that is bounded in H 1 , yet the maximal operator associated with the Bochner-Riesz sum at the critical index is greater than c log N on a set of measure at least d, where c and d are positive constants. The major elements in Stein's construction are: ( 1 ) sharp asymptotic estimates for the summability kernels, ( 2 ) number theoretic estimates that take advantage of the structure of the kernels, and ( 3 ) an intricate way of putting it all together. All of (1) and ( 2 ) and most o f ( 3 ) works, almost without variation, on En. A trivial change is that on En it is convenient to use Ceshro summa bility. That part of Stein's argument which can be dealt with by a wave of the hand (on Rn) is the only part that creates special difficulty on En. Consider three families of functions: {fR}, { g R l , {aR}, R > 1 defined on Rn and all radial. Suppose that their respective Fourier transforms are Cm, have compact support and satisfy the following conditions:
*
This research was supported by NSF grant MCS 7903 122. 245
246 M.H. Taibleson
One needs to know that { g R } is bounded in L 1 , that {fRl is bounded in H1 and that the {a,) satisfy a size condition which implies that the maximal function supR l l a R * g ( x ) l is dominated by a multiple of the Hardy-Littlewood maximal function. On Rn this is done by taking f l , g , , and u l as they come and then setting
,
fR(x)
=
R" f l ( R x ) , gR(x)
=
R" gl(Rx), a,(x)
=
R" al(Rx).
The remainder of this paper is concerned with the construction of such functions on Cn, the n-dimensional sphere in Rn+l 3 as well as on other compact rank 1 symmetric spaces. When we consider Cn we always assume that n L 2 . (See [S] where the implications o f the existence of an h in H1 with divergent means at the criti cal index is discussed). a > 0 let {Cp)} be the sequence of Gegenbauer (ultraspherical) polynomials. The sequence {Cp)} is an orthogonalization of the sequence of monomials {tk);=o on [ - 1 ,1] with respect For
to measure ( 1 - t2)(a-("2)) It is usual to set cos e manic of degree k on C (41
dt, = t, 0 is the
normalized so C p ) ( l ) = ( k + 2 a - 1 ) . <- e < T . The zonal spherical harfunction
(k+a) C p ) (cos e ) ,
u
=
(n-l)/Z.
The kernel on Cn that induces the multiplier IPk} "spherical convolution" is given by the distribution:
under
Good references on these matters are the paper of Bonami and Clerc [3] and the treatise of SzegB [7].
Gegenbauer and Jacobi Polynomials 247
Let us define: A o pk
=
pk, Alpk
- pk+2, A s + 1 k' --
= pk
(Aspk), s = 1,2,. . . The estimate of primary interest in this paper is the conclusion of the following theorem: = A
1
Theorem. Suppose
R
i s a p o s i t i v e i n t e g e r and
is a s e q u e n c e
{pk)
of r e a l numbers w i t h
(6) pk = 0 i-f
&
k > R,
e
L
( 7 1 I A pkl 5 A R-',
=
o,i
,...,
[(n+3)/21.
3 M
is d e f i n e d by 1 5 1 w i t h a = (n-1)/2 t h e r e i s a c o n s t a n t ( d e p e n d i n g o n l y on nl s u c h t h a t
(a)
IM(COS
ell
(b)
Lf
i s even
n
(A
IM(COS (c)
If
5 A B, R",
ell 5 n
A
Bn
n 2 2.
Bn R-1/2,-(n+1/2),
0 <
Bn R-1/2(n-e) - (n+l/2)
,
e 5 n/2
s/Z
5 e
< n
i s odd
Think now of (l), (2), and (3) as describing conditions on sequences {lR(k)l, {iR(k)l, and IGR(k)l. The simplest way to cons truct sequences that satisfy ( 6 ) and ( 7 ) is to start with a function J, that is Con [o,-1, supported on [o,I] and set !.ik= = J,(k/R). However, for the specific application we have in mind we need to consider a version of GR(k) for which, instead of (3), we have
+R?]
[[R-kR:k?]/[R (8)
GR(k)
=
0
,
0
5 k 5 [(R+1)/2]
, kL2R
That is, for the smaller values of k, GR(k) is the coefficient for Cesaro summability at the critical index. It is easy to show that can be constructed to satisfy ( 8 ) and (7) uniformly in R. {2R(k)) The estimates of the theorem show (via typical computations) that if h is integrable on En and aR(cos 8) = CGR(k)(k+a) C p ) ( c o s 0)
248 M.H. Taibleson
then
where 2 is the antipode to x , h* is the Hardy-Littlewood maximal function and D is a positive constant. Clearly we may construct the families Ig,) and IfR) to satisfy (2) and (1) as well as ( 7 ) . It is an immediate consequence of the estimates of the theorem that {g,) is a bounded sequence in L1 (and that If,) is a bounded sequence in L’). To complet the program we will show that {f,) is bounded in H 1 The point is 1 that {f,} is a bounded family of H -molecules in the sense of [8]. The idea behind such molecules is that they have a canonical decomposition in H 1 -atoms. The argument o f [8] is in the setting of Rn and the disk in C , but the argument for En is essentially the same. It is only necessary to allow for growth at the antipode of the center of the molecule, as well as to allow for exceptional atoms. Atomic characterizations of the HD-spaces are discussed by = 0 one sees that f z fR(U)dv(u) = 0. Colzani [4]. Since ?,(O) n The estimates of the theorem show that IfR) is a uniformly bounded sequence of (1, a, 0, 1/2n)-molecules.
.
The theorem follows from a proposition that is a direct consequence of an ohservation first made by Askey and Wainger [2] to obtain estimates f o r expansions of the form (10)
1 k-@b(k)(k+a)
CL’),
6 > 0
and
b
slowly varying.
Before procedding we will gather some facts about orthogonal polynomials that we will need below. The sequence of Jacobi polynomials
{Pi”’)),
6, B > -
:,
is an
orthogonalization of the sequence of monomials {tk}i=o on [-1 , 1 ] 6 with respect to the measure (1-t) (l+t)’ dt, normalized s o Pi6,’)(1) = Up to a multiplicative constant the Gegenbauer polynomials are special cases of Jacobi polynomials:
(’:;“I.
See [ l ] 22.5.20. From Bonami and Clerc [3] p . 231, we have the estimates:
Gegenbauer and Jacobi Polynomials 249
where
is a positive constant that does not depend on
B
k or
8.
The ratio of gamma functions in (11) is on the order of (k+l)(a-1/2) as k + -. From (11) and ( 1 2 ) follows:
I
(B( k+l)
(13)
where B
((k+a) Ci(cos
€I)(
5 B(k+l)"O-a B(k+l)a(n-e)-a,
, oce51~ , 0 < e 5 IT/^, ~ / 5 2 8 <
is a positive constant that does not depend on
( k + a ) Cia'
- 'k-(a+1)), 1
=
k or
8.
22.7.23 we have
From Abramowitz and Stegun [l] (14)
IT
(C_'!)
I
C(B) s 0 all8) -1
For convenience let us assume that {pkj has only a finite number of non-zero terms. A consequence of (14) for s = 1,2,3, ... is (15)
& pk(k+a)
cia' 1
= a
Proposition A . S u p p o s e {pk]
polynomials a n d
(17)
ck( a + l ) A~k)))(k+(a+s))Ck
(a+s)9
a > 0, I C E ) is t h e s e q u e n c e of G e g e n b a u e r is a finite sequence. T h e n
Ba
(pk((k+1)2a,
...
pk(k+a) Ck( a ) ( c o s 0 ) S
L B
Ci(Ii1))
.
s = 1,2,3,
1;
-
(A(...A(k+(a+lT1
(a+s-1)
cia'I 5
(16) (EPk(k+a)
1 l,Ik(Cia+')
APk(k+(a+l))
...
( a ) s = a(a+l)
and f o r
a
1-k 1
=
where
=
1
I
250 M.H. Taibleson
The proposition is an immediate consequence of (15) and (13). Proof of the Theorem. In Proposition A we set s = [(n-1)/2] + 2, and recall that uk = 0 if lukl 5 A we have from (16), IM(COS e l l 5 A B,
a = (n-l)/Z, k > R. Since R c (k+i)"-l 5 A B,R". k=O Examine the exponent a-s+L in (17). It is equal to: (n-1)/2 - [(n-1)/2] + L-2, which is L-(3/2) if n is even and is (L-2) if n is odd. Since L 2 1 we see that if n is even o r n is odd and L # 1 , then a-s+L > -1. Suppose now that 0 0 5 n/2. We have IAL pkl 5 A R-', Auk = 0 if k > R, e = 1,2,3 ,..., O r s. If n is even, o r n is odd and L # 1, we have the estimate: R R 1 A R-L (k+l)a-s+L 0 -(a+s) -< A Bn R-L 0 - ( a + s ) 1 (k+l)cL-s+L Bn k=O k=O R-L e-(a+s)Ra-s+L+l - A Ra-s+l O - ( a + s ) < A Bn n = [
A
B,
A
Bn R-l 0 - (n+l)
e-(n+1/2),
,
n
even
n
odd
For the exceptional case we have the estimate:
< A
-
Bn R - l e-(n+l)[l
+
log+(RB)]
This establishes the theorem for 0 5 0 5 n/2. follows froman analogous argument.
The case n/2 5 0 n'
We now consider an analogue of Proposition A for Jacobi polynomials. There are several reasons why this is of interest. First, the Jacobi polynomials contain the Gegenbauer polynomials as special cases (see (11)). Second, the zonal functions for all compact symmetric spaces of rank 1 are obtained as special cases of Jacobi polynomials: En
and
Pn(R),
n
2,
;
Pn(C),
n 2 4,
Gegenbauer and Jacobi Polynomials 251
{
n
n 2 8, {Pi-'
Pn(E),
';P
"};
a n d t h e e x c e p t i o n a l case,
{F'L'93)1. S e e [3] p . 257. T h i r d , e v e n f o r t h e s p e c i a l case o f t h e e s t i m a t e s f o r t h e "lower h e m i s p h e r e " c a n b e improved s i g n i 'n9 f i c a n t l y . I n c o m p a r i s o n w i t h ( 9 ) we c a n o b t a i n , u s i n g P r o p o s i t i o n B y below: '16'
(18)
laR*h(x)l <
(19)
l i m supR
D(h*(x) + R -
- {D(h*(x)
,
-
laR
*
+
1/2h * (G)),
n
even
R-l l o g R h * ( G ) ) ,
n
odd
h(x)I 5 D h*(x).
The expansions o f i n t e r e s t ( i n a n a l o g y w i t h (5))
a r e of t h e
form:
where
22.7.18 we h a v e
From Abramowitz a n d S t e g u n [l]
(22) where
(2k+6+6+1) for a l l
PLYyB) ! 0
=
6,B.
From (21) a n d (22) we h a v e (23)
,(6,B)
k
k'
(698) =
(k+6+6+1) Pk(6+19 B )
-
(k+B) Pk-l ( 6 + 1 Y B)
,
252 M.H. Taibleson
1 Proposition B. S u p p o s e 6,B > -7, {Pi6")} is t h e s e q u e n c e of J a c o b i p o l y n o m i a l s and {pk} i s a f i n i t e s e q u e n c e . Then
and for s
=
1,2,3,...
0-(6+s+1/2)
o
e
<
~ / 2
Proposition B is a direct consequence of (24) and ( 2 5 ) . The estimates for the kernels on En that are obtained from - 1 are the same as those obtained Proposition B with 6 = B = from Proposition A on the upper hemisphere (i.e., 0 5 0 5 n/2). < n) there is a substantial imFor the lower hemisphere (n/2 2 0 provement. (The only difference in the argument is to use (27) for both parts of the estimate). A comparison of the estimates obtained from the two propositions follows:
4
From Proposition A ,
~ / 52 0
i T
B, min {R", R - ~( */- e~) - (n+1/2) I ,
n
even
IM(cos 011 5 Bn min {Rn, R-'(~r-0)-(~'')(1
+ log+fR(n-0))},
n odd
Gegenbauer and Jacobi Polynomials 255
Re ferences [l]
M . Abramowitz and I. Stegun, Handbook of Mathematical Functions,
Dover, New York, 1 9 7 0 . [2]
R. Askey and S. Wainger, On the behaviour of special classes of ultraspherical expansions, I., Journ. d. Anal. Math. 1 5 ( 1 9 6 5 ) 193-220.
[3]
A. Bonami and J . L. Clerc, Sommes de Cesaro et multiplicateurs des developpments en harmonique spheriques, Trans. Amer. Math. SOC., 1 8 3 ( 1 9 7 3 ) 2 2 3 - 2 6 3 .
[S]
L. Colzani, Hardy and Lipschitz spaces on unit sphere, Ph. D. dissertation, Washington University, St. Louis, MO, 1 9 8 2 .
[S]
L. Colzani, M . Taibleson, and G. Weiss, Cesaro and Riesz means on the unit sphere. To appear.
[6]
E. M. Stein, An H 1 function with non-summable Fourier expansion, to appear in Proc. of Cortona Conf. on harmonic analysis, 1982.
[7]
[8]
G . Zseg:,
Orthogonal Polynomials, Amer. Math. SOC. Colloq. Pub. No. 2 3 , 4th ed. Providence, RI, 1 9 7 5 .
M. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Asterisque 77 (1 9 8 0 ) 6 7 - 1 4 9 .
ESTIMATES FOR FINITE EXPANSIONS OF GEGENBAUER AND JACOB1 POLYNOMIALS Mitchell H. Taibleson* Washington University in St. Louis St. Louis, Missouri 63130
Estimates will be given for certain Gegenbauer polynomial and Jacobi polynomial expansions. The primary interest is on expansions of a form that are needed to extend a result of E.M. Stein [6]. Stein shows that there is a function h E H 1 (Rn) (resp., h E H 1 (Tn1) such that the Bochner-Riesz means of h at the critical index (n-1)/2 diverge a.e. He accomplishes this as a consequence of the construction of a sequence {hN) that is bounded in H 1 , yet the maximal operator associated with the Bochner-Riesz sum at the critical index is greater than c log N on a set of measure at least d, where c and d are positive constants. The major elements in Stein's construction are: ( 1 ) sharp asymptotic estimates for the summability kernels, ( 2 ) number theoretic estimates that take advantage of the structure of the kernels, and ( 3 ) an intricate way of putting it all together. All of (1) and ( 2 ) and most o f ( 3 ) works, almost without variation, on En. A trivial change is that on En it is convenient to use Ceshro summa bility. That part of Stein's argument which can be dealt with by a wave of the hand (on Rn) is the only part that creates special difficulty on En. Consider three families of functions: {fR}, { g R l , {aR}, R > 1 defined on Rn and all radial. Suppose that their respective Fourier transforms are Cm, have compact support and satisfy the following conditions:
*
This research was supported by NSF grant MCS 7903 122. 245
246 M.H. Taibleson
One needs to know that { g R } is bounded in L 1 , that {fRl is bounded in H1 and that the {a,) satisfy a size condition which implies that the maximal function supR l l a R * g ( x ) l is dominated by a multiple of the Hardy-Littlewood maximal function. On Rn this is done by taking f l , g , , and u l as they come and then setting
,
fR(x)
=
R" f l ( R x ) , gR(x)
=
R" gl(Rx), a,(x)
=
R" al(Rx).
The remainder of this paper is concerned with the construction of such functions on Cn, the n-dimensional sphere in Rn+l 3 as well as on other compact rank 1 symmetric spaces. When we consider Cn we always assume that n L 2 . (See [S] where the implications o f the existence of an h in H1 with divergent means at the criti cal index is discussed). a > 0 let {Cp)} be the sequence of Gegenbauer (ultraspherical) polynomials. The sequence {Cp)} is an orthogonalization of the sequence of monomials {tk);=o on [ - 1 ,1] with respect For
to measure ( 1 - t2)(a-("2)) It is usual to set cos e manic of degree k on C (41
dt, = t, 0 is the
normalized so C p ) ( l ) = ( k + 2 a - 1 ) . <- e < T . The zonal spherical harfunction
(k+a) C p ) (cos e ) ,
u
=
(n-l)/Z.
The kernel on Cn that induces the multiplier IPk} "spherical convolution" is given by the distribution:
under
Good references on these matters are the paper of Bonami and Clerc [3] and the treatise of SzegB [7].
Gegenbauer and Jacobi Polynomials 247
Let us define: A o pk
=
pk, Alpk
- pk+2, A s + 1 k' --
= pk
(Aspk), s = 1,2,. . . The estimate of primary interest in this paper is the conclusion of the following theorem: = A
1
Theorem. Suppose
R
i s a p o s i t i v e i n t e g e r and
is a s e q u e n c e
{pk)
of r e a l numbers w i t h
(6) pk = 0 i-f
&
k > R,
e
L
( 7 1 I A pkl 5 A R-',
=
o,i
,...,
[(n+3)/21.
3 M
is d e f i n e d by 1 5 1 w i t h a = (n-1)/2 t h e r e i s a c o n s t a n t ( d e p e n d i n g o n l y on nl s u c h t h a t
(a)
IM(COS
ell
(b)
Lf
i s even
n
(A
IM(COS (c)
If
5 A B, R",
ell 5 n
A
Bn
n 2 2.
Bn R-1/2,-(n+1/2),
0 <
Bn R-1/2(n-e) - (n+l/2)
,
e 5 n/2
s/Z
5 e
< n
i s odd
Think now of (l), (2), and (3) as describing conditions on sequences {lR(k)l, {iR(k)l, and IGR(k)l. The simplest way to cons truct sequences that satisfy ( 6 ) and ( 7 ) is to start with a function J, that is Con [o,-1, supported on [o,I] and set !.ik= = J,(k/R). However, for the specific application we have in mind we need to consider a version of GR(k) for which, instead of (3), we have
+R?]
[[R-kR:k?]/[R (8)
GR(k)
=
0
,
0
5 k 5 [(R+1)/2]
, kL2R
That is, for the smaller values of k, GR(k) is the coefficient for Cesaro summability at the critical index. It is easy to show that can be constructed to satisfy ( 8 ) and (7) uniformly in R. {2R(k)) The estimates of the theorem show (via typical computations) that if h is integrable on En and aR(cos 8) = CGR(k)(k+a) C p ) ( c o s 0)
248 M.H. Taibleson
then
where 2 is the antipode to x , h* is the Hardy-Littlewood maximal function and D is a positive constant. Clearly we may construct the families Ig,) and IfR) to satisfy (2) and (1) as well as ( 7 ) . It is an immediate consequence of the estimates of the theorem that {g,) is a bounded sequence in L1 (and that If,) is a bounded sequence in L’). To complet the program we will show that {f,) is bounded in H 1 The point is 1 that {f,} is a bounded family of H -molecules in the sense of [8]. The idea behind such molecules is that they have a canonical decomposition in H 1 -atoms. The argument o f [8] is in the setting of Rn and the disk in C , but the argument for En is essentially the same. It is only necessary to allow for growth at the antipode of the center of the molecule, as well as to allow for exceptional atoms. Atomic characterizations of the HD-spaces are discussed by = 0 one sees that f z fR(U)dv(u) = 0. Colzani [4]. Since ?,(O) n The estimates of the theorem show that IfR) is a uniformly bounded sequence of (1, a, 0, 1/2n)-molecules.
.
The theorem follows from a proposition that is a direct consequence of an ohservation first made by Askey and Wainger [2] to obtain estimates f o r expansions of the form (10)
1 k-@b(k)(k+a)
CL’),
6 > 0
and
b
slowly varying.
Before procedding we will gather some facts about orthogonal polynomials that we will need below. The sequence of Jacobi polynomials
{Pi”’)),
6, B > -
:,
is an
orthogonalization of the sequence of monomials {tk}i=o on [-1 , 1 ] 6 with respect to the measure (1-t) (l+t)’ dt, normalized s o Pi6,’)(1) = Up to a multiplicative constant the Gegenbauer polynomials are special cases of Jacobi polynomials:
(’:;“I.
See [ l ] 22.5.20. From Bonami and Clerc [3] p . 231, we have the estimates:
Gegenbauer and Jacobi Polynomials 249
where
is a positive constant that does not depend on
B
k or
8.
The ratio of gamma functions in (11) is on the order of (k+l)(a-1/2) as k + -. From (11) and ( 1 2 ) follows:
I
(B( k+l)
(13)
where B
((k+a) Ci(cos
€I)(
5 B(k+l)"O-a B(k+l)a(n-e)-a,
, oce51~ , 0 < e 5 IT/^, ~ / 5 2 8 <
is a positive constant that does not depend on
( k + a ) Cia'
- 'k-(a+1)), 1
=
k or
8.
22.7.23 we have
From Abramowitz and Stegun [l] (14)
IT
(C_'!)
I
C(B) s 0 all8) -1
For convenience let us assume that {pkj has only a finite number of non-zero terms. A consequence of (14) for s = 1,2,3, ... is (15)
& pk(k+a)
cia' 1
= a
Proposition A . S u p p o s e {pk]
polynomials a n d
(17)
ck( a + l ) A~k)))(k+(a+s))Ck
(a+s)9
a > 0, I C E ) is t h e s e q u e n c e of G e g e n b a u e r is a finite sequence. T h e n
Ba
(pk((k+1)2a,
...
pk(k+a) Ck( a ) ( c o s 0 ) S
L B
Ci(Ii1))
.
s = 1,2,3,
1;
-
(A(...A(k+(a+lT1
(a+s-1)
cia'I 5
(16) (EPk(k+a)
1 l,Ik(Cia+')
APk(k+(a+l))
...
( a ) s = a(a+l)
and f o r
a
1-k 1
=
where
=
1
I
250 M.H. Taibleson
The proposition is an immediate consequence of (15) and (13). Proof of the Theorem. In Proposition A we set s = [(n-1)/2] + 2, and recall that uk = 0 if lukl 5 A we have from (16), IM(COS e l l 5 A B,
a = (n-l)/Z, k > R. Since R c (k+i)"-l 5 A B,R". k=O Examine the exponent a-s+L in (17). It is equal to: (n-1)/2 - [(n-1)/2] + L-2, which is L-(3/2) if n is even and is (L-2) if n is odd. Since L 2 1 we see that if n is even o r n is odd and L # 1 , then a-s+L > -1. Suppose now that 0 0 5 n/2. We have IAL pkl 5 A R-', Auk = 0 if k > R, e = 1,2,3 ,..., O r s. If n is even, o r n is odd and L # 1, we have the estimate: R R 1 A R-L (k+l)a-s+L 0 -(a+s) -< A Bn R-L 0 - ( a + s ) 1 (k+l)cL-s+L Bn k=O k=O R-L e-(a+s)Ra-s+L+l - A Ra-s+l O - ( a + s ) < A Bn n = [
A
B,
A
Bn R-l 0 - (n+l)
e-(n+1/2),
,
n
even
n
odd
For the exceptional case we have the estimate:
< A
-
Bn R - l e-(n+l)[l
+
log+(RB)]
This establishes the theorem for 0 5 0 5 n/2. follows froman analogous argument.
The case n/2 5 0 n'
We now consider an analogue of Proposition A for Jacobi polynomials. There are several reasons why this is of interest. First, the Jacobi polynomials contain the Gegenbauer polynomials as special cases (see (11)). Second, the zonal functions for all compact symmetric spaces of rank 1 are obtained as special cases of Jacobi polynomials: En
and
Pn(R),
n
2,
;
Pn(C),
n 2 4,
Gegenbauer and Jacobi Polynomials 251
{
n
n 2 8, {Pi-'
Pn(E),
';P
"};
a n d t h e e x c e p t i o n a l case,
{F'L'93)1. S e e [3] p . 257. T h i r d , e v e n f o r t h e s p e c i a l case o f t h e e s t i m a t e s f o r t h e "lower h e m i s p h e r e " c a n b e improved s i g n i 'n9 f i c a n t l y . I n c o m p a r i s o n w i t h ( 9 ) we c a n o b t a i n , u s i n g P r o p o s i t i o n B y below: '16'
(18)
laR*h(x)l <
(19)
l i m supR
D(h*(x) + R -
- {D(h*(x)
,
-
laR
*
+
1/2h * (G)),
n
even
R-l l o g R h * ( G ) ) ,
n
odd
h(x)I 5 D h*(x).
The expansions o f i n t e r e s t ( i n a n a l o g y w i t h (5))
a r e of t h e
form:
where
22.7.18 we h a v e
From Abramowitz a n d S t e g u n [l]
(22) where
(2k+6+6+1) for a l l
PLYyB) ! 0
=
6,B.
From (21) a n d (22) we h a v e (23)
,(6,B)
k
k'
(698) =
(k+6+6+1) Pk(6+19 B )
-
(k+B) Pk-l ( 6 + 1 Y B)
,
252 M.H. Taibleson
1 Proposition B. S u p p o s e 6,B > -7, {Pi6")} is t h e s e q u e n c e of J a c o b i p o l y n o m i a l s and {pk} i s a f i n i t e s e q u e n c e . Then
and for s
=
1,2,3,...
0-(6+s+1/2)
o
e
<
~ / 2
Proposition B is a direct consequence of (24) and ( 2 5 ) . The estimates for the kernels on En that are obtained from - 1 are the same as those obtained Proposition B with 6 = B = from Proposition A on the upper hemisphere (i.e., 0 5 0 5 n/2). < n) there is a substantial imFor the lower hemisphere (n/2 2 0 provement. (The only difference in the argument is to use (27) for both parts of the estimate). A comparison of the estimates obtained from the two propositions follows:
4
From Proposition A ,
~ / 52 0
i T
B, min {R", R - ~( */- e~) - (n+1/2) I ,
n
even
IM(cos 011 5 Bn min {Rn, R-'(~r-0)-(~'')(1
+ log+fR(n-0))},
n odd
Gegenbauer and Jacobi Polynomials 255
Re ferences [l]
M . Abramowitz and I. Stegun, Handbook of Mathematical Functions,
Dover, New York, 1 9 7 0 . [2]
R. Askey and S. Wainger, On the behaviour of special classes of ultraspherical expansions, I., Journ. d. Anal. Math. 1 5 ( 1 9 6 5 ) 193-220.
[3]
A. Bonami and J . L. Clerc, Sommes de Cesaro et multiplicateurs des developpments en harmonique spheriques, Trans. Amer. Math. SOC., 1 8 3 ( 1 9 7 3 ) 2 2 3 - 2 6 3 .
[S]
L. Colzani, Hardy and Lipschitz spaces on unit sphere, Ph. D. dissertation, Washington University, St. Louis, MO, 1 9 8 2 .
[S]
L. Colzani, M . Taibleson, and G. Weiss, Cesaro and Riesz means on the unit sphere. To appear.
[6]
E. M. Stein, An H 1 function with non-summable Fourier expansion, to appear in Proc. of Cortona Conf. on harmonic analysis, 1982.
[7]
[8]
G . Zseg:,
Orthogonal Polynomials, Amer. Math. SOC. Colloq. Pub. No. 2 3 , 4th ed. Providence, RI, 1 9 7 5 .
M. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Asterisque 77 (1 9 8 0 ) 6 7 - 1 4 9 .