Bateman's equation and similar units

NOKIH-HOILAND

Bateman’s Friedrich

Equation

and Similar Units

Roesler

Mathemutisches lnstitut der Technischen Universittit Miinchen 80333 Miinchen, Germany

Submitted by Richard A. Rruakli

ABSTRACT

The multiplication K(r, y)o F( y, z) = jK(x, y)F( y, z) dy of real functions K and F can be interpreted as the analytic version of matrix multiplication. This suggests examining

whether

that E(x,

y)o F(y,z)

independent such

1.

E(x,

f. Bateman’s y).

and similar units.

has a unit element,

i.e., a kernel

z ) or lE( x, y )f( y ) dy = f(x)

= F(x,

functions

a kernel

function

this multiplication

This

function

paper

[sin(x

develops

0 Elsevier

-

y)]/rr(

a procedure

Science

for infinitely

E(x,

y) such

many linear

x - y) is an example to construct

of

Bateman’s

Inc., 1997

INTRODUCTION

The operator

f(x) + can be interpreted variables, i.e.,

LINEAR

ALGEBRA

lK(x, YMY)

as a multiplication

on spaces of functions

AND ITS APPLICATIONS

0 Elsevier Science Inc., 1997 F~55Avenue of the Americas, New York, NY 10010

dY

250:253-273

in two real

(1997) 0024-3795/97/$17.00 SSDI 0024-3795(95)00528-Y

254

FRIEDRICH

and hence appears as the analytic analogy suggests examining whether i.e., a kernel E(x, Y) such that

ROESLER

version of matrix multiplication. This this multiplication has a unit element,

qx,y)oqy,z)

=F(x,z)

(1.2)

=f(x)

(1.3)

or

jw

dY

Y)f(Y)

I

for infinitely

many linear independent

Bateman’s

equation

[l,

functions

f.

p. 483, formula (38)]

m sin(x-y) _-m 7T(X _ y) f(Y) /

(I-4)

dY =f(r)

provides an example of such a kernel E( x, y). G. H. Hardy commented this equation in his introduction of [2] as follows: In one of his papers on integral equations, of the

equation

f(x)

Mr. H. Bateman

= (I/~)l~&in(t

-

striking in itself and capable of interesting

x)/(t

The main subject of [2] is to determine

a formula

E(x,

y)

by the

We start with the initial part [viewed as an R(x)~

FN(x)

of an appropriate

sequence

:=

which

i.e.

following

column vector]

(frdx))l
of real functions F(x)

:=

(f”WL

by the functions f,,,
MN

:=

is

... .

extensive classes of “l-functions,”

functions f which satisfy Equation (1.4). In this paper we construct unit elements

procedure:

has stated and made use

- x>)f(t)dt,

applications

on

(k:)n)l
:=

y), for

(l-5)

BATEMAN’S

EQUATION

AND SIMILAR

is assumed to be positive definite.

denotes

Then the unit element

the inverse of M,.

The intervals I, c I, c I, c

***

are chosen such that all limits

and hence E := (%,“L”>l exist. Then

is a candidate

for a kernel in Equation

I:=

(1.3) with

UZN. N>l

The uniquely determined

upper Cholesky

factorization

E, = H;H, of E,

as a product

of an upper triangular

255

UNITS

matrix

of A,

is

256

FRIEDRICH

with positive diagonal elements,

and its transpose

an appropriate orthonormal system of functions tation of E,(x, y) defined in (1.6), i.e.,

The system is appropriate

ROESLER

Hi, yields via

for an orthonormal

represen-

in the sense that all limits

77,,L

:= 1,

lim n!;“), N+ -a

and hence

exist. Therefore the orthonormal the limit case N = 00:

functions

I+!I(“) can also be carried over to ,,I

This suggests the representation

E(r,

y) =

(1.8)

as the limit case of Equation (1.7). Starting with f,,
= x”-’

and I,

= [-N,

N] yields

kernel

E(x,

y) =

sin( x - y)

7i-(x-y)

’

BATEMAN’S

EQUATION

AND SIMILAR

(2) the orthonormal

257

UNITS

system

where J”(X) denotes the Bessel function of order v, and (3) the addition theorem for the spherical functions

which is a special case of a formula of Clebsch [6, p. 363, formula (3)] and by which Bateman was originally led to his equation (see [2, p. 4471). In Section 2 the general procedure in the finite-dimensional case is developed: the trick is to orthonormalize the system
We refrain from a discussion of which functions F or f actually satisfy Equation (1.2) or (1.3), and will be content to repeat some known results concerning Bateman’s original equation (1.4). Further

notation:

For matrices

A4 = (v ,,,, n)l ~ ,,, n 4 N,

if

(Y,Jj
M(j,k) := 1

if

j
1

if

j=r=s=k,

0

otherwise.

( v,,,n ) , c rn M( j, k; r, s) :=

l
otherwise,

G k

rn

z

r

j$fi
Ii

(-l)"*+"lM(l,m;n,m)I

L, :=

IM(l,m - l)IIM(lTm)l

i

IG,n,nSN

(1.9)

258

FRIEDRICH

with M = M,

a lower triangular

HN := (viy;)

matrix with positive diagonal elements,

N)llM(m

S”,_ ” :=

S“L< 11:=

AND

1

if

0

otherwise,

1 0

if m
l
+ I, N)I

with M = M, an upper triangular matrix with positive s m,n denotes the Kronecker symbol, and

FORWARD

and

(-l)“+“~M(m,N;n,m)(

:=

IM(m,

2.

ROESLER

diagonal

elements.

m =n(mod2),

BACKWARD

ORTHONORMALIZATION

PROPOSITION1. (1) Forward orthonormalization I, + R defined by

(If ( fl, . . . , fN):

The functions

4’)

:

The functions

I+!&~):

(&y’(+,<,<-. . , := LPN(4 are orthonormal: j,,v +kN ‘( x ) 4: M‘( x ) dx = S,,&,n. (2) Backward orthonormulisation of (f, , . . . , f,): I, + R defined by

II,,,I,!J!~” ‘( x 1I,!(N ‘( x 1 dx = S ,,,, n.

are orthonormal: Proof.

4~:“‘(~)

(1): By the definition

=

c 1 cj

(._l)l~l+j < 711

of L,

in Equation

(1.91,

lM(1,m;j, m)l dlM(l,m - l)I/M(l,m)l

fjw-

BATEMAN’S

EQUATION

AND SIMILAR

UNITS

259

Hence for 1 < m < n =GN

\/IM(l,m- l)lIM(l,m)I \/IM(l,n - l>IIM(l,n)I

=

/( 1,

X

(

c

C

l
=

(-l)“‘+‘~M(l,

m:i,m)b(x))

l$i
(-l)~“l~(l,,i:i,n)~(x)) dx

l~~,~(-l)~“+ilM(l,m;i,m)I . . x

C (-l)“+‘~~;\“IM(l,n;j,n)l i l
(N)

Pll =

C

(-l)“‘+‘IM(l,m;i,m)I*

l
,n

-

l)IIM(l,n)I

I-Y-

I

PY’

‘: . . .

and, since the last column of the p-determinant l
iyiq1

...

! /Q

=

bY (1.5)

1

(N) P n,n-I

is duplicated

if

m
if

m =n.

: CL(N) nr

for each i,

(2): Orthonormalize forward according to Proposition and denote the resulting system by (I&$“‘, . . . , (CIiN’>.

l(l), n

REMARKS.

(1) ($qN’, . . . , I,!+$“‘)is a maximal orthonormal system in A,;

hence by

FRIEDRICH

260 Proposition

l(2)

A comparison

with the representation

which is the upper Cholesky

EN(X,Y) = according

ROESLER

1$

of E,(x,

factorization

of E,.

c iv 4tY44T’(Y)

=

(1.6) shows

Similarly

~NW’GJLNFN(Y)

m Q

to Proposition

(2) Orthonormalization

l(l),

which gives the lower Cholesky

forwards according to Proposition l(1) (*I’N’ >.a., I,!J-$~)) as orthonormalization to Proposition l(2). That is,

L’,G,(x) where L!, denotes the matrix L, M = M,, for here we have

(2.2)

factorization

of the functions

G.v(x) := (d;“‘(x)>, <,,,<,v

To verify Equation

y) in Equation

:=

ENFN( x)

yields the same orthonormal system of (f, , . . . , fN) backwards according

= (@,!%>),.,n..~ in Equation

it is sufficient

(2-l)

(2.2)

(1.9) with M = EN instead of

to observe

that by Equation

(2.1)

LJNGN( x) = L!, E, FY( x> and that-like H,--L:, E, is an upper triangular matrix with positive diagonal elements. Both matrices L!NE,Y and H, transform (f,, . . . , _fs 1 into an orthonormal system; hence they must be equal.

BATEMAN’S

EQUATION

The representation

AND SIMILAR

UNITS

(2.2) of the functions

261

I,!J:~“’ in terms of (gi”),

. . . , gf;“‘>

with the coefficients of L’, is sometimes more favorable than that of . . . , f,,,) with the coefficients of H,--see the Proposition l(2) in terms of example f,!(r) = e-“‘” in Section 4.


3.

TWO

CONCRETE

We first establish

CASES some necessary

technical

results.

2.

PROlTSITlON

(2) The iwersc

of the

~

matrix A/i =

M-‘=

1 < m. ?1 < N

is

A,,,A,,

( i ~

n,,,+ a,,

wifh A,,, =

n

I

(a,,, + a.,)

I
I-I (% - q.

IGj9.V

(3) The inverse

Proof. (1) IS a tl leorem consequences of (1).

of Cauchy

[3, p. 871. (2) and (3) are immediate w

262

FRIEDRICH REMARK.

Suppose

that 0 < a, < n2 < ... . Then,

referring

ROESLER

to Proposi-

tion 2 above,

A,,,4

= (-

l)ff’-“IA,,,A,I

in part (2), and

A,,,A,

= ( -1)‘“‘-““21A,,,A,I

if

m = n(mod2)

in part (3). CASE 1.

The functions

(fn( x)), ~ , and the intervals

I,

are chosen

such

that for all N

with 0 < a, < a2 <

. . . and c,,,(N)

> 0.

With the notation

,(‘Y)

,,I

Proposition

:=

A( N, ~11) c,,,(N)

’

1 < nz < N,

and

CX,,,:=

lim cuiy’,

N+ =

(3.2)

2 shows

(3.3)

BATEMAN’S

EQUATION

E,

the coefficients

contains

ing to Equation

(1.6).

AND SIMILAR

UNITS

263

eLNi in the representation of E,(x, y) accordcontains the coefficients of the functions

HN [ L’i]

(CICN) in terms of fi, . . . ,_fN [gj”), . . . , gkN)] according to Proposition m [Equation (2.211. Finally, the functions giN) are normalized to

(-1)”

hc,N’(x) := Fgp(

m

l(2)

l
x),

i.e. such that

/ Then the equations

PROPOSITION 3.

with

the orthononml

hc,N’(X)hc,N’(x) dx = 1,

(3.6) ”

(3.31, (3.4, Under

1 a,,.

(3.51, (2.1), and (2.2) imply:

Case 1,

functions

264

FRIEDRICH

ROESLER

and

hiNN’( x) =

c

l)‘rL&L.(x).

(-

l
The corresponding

limit-case formulae

are

and h,,(x)

=

c (-l)mz 111 >I

a,, + anfnz(‘)’

CASE 2. The functions (f,,(x)), that for all N

a 1 and the intervals

with 0 < a, < a2 <

> 0.

... and c,,,(N)

I,

are chosen such

Then with the modified notation

A(n,m)

:=

n

(a,,, + aj)

n

I
I

la, -

ajl,

(3.1’)

l
and again with

a(N) In

.= .

A(N,m) c,,(N)

’

1 G m G N,

and

LX,,,:=

lim cykN”, (3.2’)

iv-m

BATEMAN’S

EQUATION

AND SIMILAR

UNITS

265

Proposition 2 shows

a!nvaw

-l)(n-m)‘2

S,,_(

’ arrr+ a,

(3.3’) 1$ ,?InGN

and hence PROPOSITION3’.

E&,(x,

Under Case 2,

c

y) =

(-l)(n~‘n)‘2

;f;r’f.(~)f~(y) n

l
with the orthonormal

The corresponding

functions

limit-case formulae

E(x, Y> =

are

c (-l)‘““““~~~(x)f~(y),

7ll,fl>l

n

m = n (2)

i)/“*(x)

=

J2a,

c n>m n=m(2)

(

-l)(n-m)‘2A(;:n)fn(x).

FRIEDRICH

266 4.

ROESLER

EXAMPLES EXAMPLE 1.

This corresponds

f,,(x)

:= x’)-~,

and

(3.1’) and (3.2’)

c,,,(N)

c > 0. Hence

n

(m+j-1)

j = rn(2)

and the Wallis product

formula yields

3’ gives

= fi(

N/c)fi’-1’2.

show that

1
Now Proposition

N/c],

to Case 2 with

n ?I, = m - i The equations

1, := [ -N/C,

I

n

Im-jl

l$j
BATEMAN’S

EQUATION

AND SIMILAR

UNITS

267

hence sinc(x E(x,

y)

=

-y)

4x-y)

and

x c (_1)(?1~111)/2 [(n + m)Pl! (Yl+ m)! [(n - m),2]!(“)n-‘. II>n, 2 m

,I = rn(2)

Using Legendre’s

duplication

formula,

In particular, the limit case Equation (1.81, becomes

it follows that

of the orthonormal

representation

(1.71,

i.e.

sin c( x - y) Tr(XFy)

= &

,?X, (m -

+)J,,,?

I,“(cQk&Y).

Concerning the orthonormalit)l of the functions I/J,,,, m > 1, see [2, p. 4481. Some examples of functions f which satisfy Bateman’s equation (1.4):

f(x)= 1, f(X)

=cosux,

f(X)

f(X)

sin ns = ~ s 1

f(x)

= x_“‘J,,(

f(x)

= PJ&),

x),

=sinnx-,

Inl<

p > -+

15, p. 3501, [2, p. 4471,

Inl < I

0 i m i

1

Jl

[5, p. 3521, [2, p. 4601.

268

FRIEDRICH

ROESLER

REMARKS. (1) Hardy’s (2) to he a

E(x, lj)” ECy, z > = E(x, 2). (See [2, p. 449, formula (411. Also see footnote on p. 451 and Section VI of [2].) A necessary and sufficient condition for a function f< x) of L2( - ~0, a> solution of Equation (1.4) is that it should he of the form

where F is L2( -1,

I)

[5, p. 350, Theorem 1561. (3) In particular,

where P,,(t) are the Legendre polynomials [4, p. 2311. Thus the functions m 2 1, with c = 1 satisfy Bateman’s equation. EXAMPLE 2.

f,>(x):= xrrLt,r,s~Z,r>l,s> I, :=

[-(N/c)“‘, (N/c)“‘]

-r,rodd,

)

c > 0.

Set 2s + 1 p:=

2r’

Then

hence this corresponds

to Case 2 with

n !?I = r( nr + p)

and

c,,,( N)

= fi(

N/c)“‘+

p.

I,!J,,,,

BATEMAN’S

EQUATION

The Equations

AND SIMILAR

(3.1’) and (3.2’)

UNITS

269

show

where r denotes the classical yields in the limit case

gamma

function.

Therefore

Proposition

3’

and

for E(x,

A closed representation rc

E(x,

y) = -

y> is (for x # y)

(~y)~-l’~

2

[.L+,(~‘)J,(4)

xr-?jr

which can be verified directly by comparing (X

- Y)

the coefficients

of the identity

c (m + P)jr,l+p(2X)J,,l+p(2y) ?ll> 1 =

REMARK.

JqJl+p(2x)Jpw)

The example fn(x) I,

corresponds result is

- I,(cr%+,(“Yr)]~

to Case

E(x, y) =

-Ipw)~,+,w)].

:= x~~“+~, r > 0, ,s > -2r,

:= [O, (2N/c)“‘])

1 and may be handled

c > 0, using a similar argument.

The

270

FRIEDRICH

and,

in closed

representation,

EXAMPLE 3.

which

f,,< x> := em”’ ‘,

corresponds

by Equations

= Z := [O, x).

zz ,$

and

cgII( N)

= 1.

(3.1) and (3.2)

Ly,,, = (‘T’li - cm”“‘). rr( Thus

with

Proposition

3 yields

This gives

1 witll

to Case 0 !,>

Hence

I,

ROESLER

BATEMAN’S

EQUATION

AND SIMILAR

UNITS

271

In addition set

h,,(x)

:=

%

C mEE,m#O

Then, for x > 0 and y > 0, it is a straightforward check that

Y) = jd;hb( x + z)ti,,(

E(x,

y + Z) dz.

Hence, with

H(x,

y) :=

-Mo(x + y),

we have the factorization E( x, Z) = H( x, The coefficients of M,

/

y)” ff( y, z).

here show that Equation (3.6) becomes

,h~‘( x)h’,N’( x)

dx =

’

m2 + 12’ ’

l
(4.2)

m,n>l.

(4.3)

The limit case of this formula is

/,ch,,(x)h,,(r)dr=

mZ:n,,

Equation (4.3) can be verified by proving for c > 0

( m2 + n2)/Yh,( c

=

x)h,(

x) dx

-hn(c)h,(c)

-

lrnh’,( x)hn(x)

dx -

j=hh( x)Mx) c

and further for n > 1,

;xr&h”(c) = 0

and

~~~~h’o(x)h,,(x)dr

= -i,

A

FRIEDRICH

272

which are applications

of the theta-function

[7, p. 4761. Now the finite-dimensional

transformation

ROESLER

formula

theory can be used to prove

(4.4 For Equation (4.1) and Proposition 3 assert that the representation of +,,, in h,,, has the same coefficients as I,!J;“’ in terms of terms of Jr,,..., hi” ) , ?tt$“. Hence the Equations (4.2) and (4.3) and the orthonormality of the functions I/I!:’ imply Equation (4.4). REMARK. compact:

This example

ffL(

x)

:=

can be modified

X+I/e

with

so that the interval

I,

Z becomes

= z := [o, 11

yields the same matrix

and hence

generates

the unit element

REFERENCES H. Baternan, On the inversion of n definite integral,

4:461-498 (1906). G. H. Hardy, On an integral equation, (1909). S. Kaczmarz 19.51.

Proc.

Proc. Lo&m

Math.

Sm.

(2)

Lonrlon Math. Sm. (2) 71445-472

and II. Steinhaus, T!leorie c1er Orthogonalreihen,

Chelsea,

New York,

BATEMAN’S 4 5

AND

SIMILAR

273

UNITS

W. Magnus,

F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, Berlin, 1966. E. C. Titchmarsh, introduction to the Theory of Fourier Integrals, Clarendon,

Oxford, 6

EQUATION

1948.

G. N. Watson,

A Treatise on the Theory of Bessel Functions,

Cambridge

U.P.,

1966. 7

E. T. Whittaker U.P.,

and G. N. Watson,

A Course

of Modern Analysis,

1973.

Receiaed 7 March 1994; final manuscript accepted 22 May 1995

Cambridge