NATIONAL BUREAU OF STANDARDS REPORT

2831

-rror bounds for eigenvalues of symmetric integral equation

by

Helmut Wielandt

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS REPORT

MBS PROJECT NBS REPORT

1 1 02-10-1 1 0*+ October 1, 1953 2831

Error bounds for eigenvalues of symmetric integral equations

by

Helmut Wielandt

American University

Universitat Tubingen, Germany

This paper has been prepcJ^^r^firaer a Standards contract with the^merican

National

Universi

U. $. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

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ty.

of

The publication, repi

unless permission iso 25.D.C, Such perm

cal ly prepared If ths

Approved for public release by the

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part, Is prohibited

idards, Washington

>rt has been speclflort for Its own use,
7 1 ?"? :? .? i: 7

7'.

, c 77 uiiT '• :v:

,; .7 ;

7 7

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* ' r4"
1 A Error bounds for eigenvalues of symmetric integral equations

by

Helmut Wielandt

1 • The problem .

When the theory of integral eauations was developed

fifty years ago, its aim was to provide a general method for

dealing with the common features of large groups of problems,

rather than to furnish an effective method to solve specific

problems numerically. Tn fact, the determination of an unknown function of one variable as the solution of an integral

equation involves a function of two variables, viz. the

kernel, and hence requires an amount of numerical work which

could not, as a rule, effectively be performed with the

equipment available at that time. Thus, the classical theory

was interested in existence theorems, convergence theorems

and asymptotic formulae more than in numerical aspects. The

situation has been changed by the recent development of high

speed calculating machines, which has created considerable

interest especially in the problem of error estimates. It

may be seen from BUCKNF.R * s report Til that much has been done

in this direction during the last years. However, interesting

1 ) This paper has been prepared under a National Bureau of

Standards contract with the American University.
-2

-

and important questions are still open, For instance, in

the basic problem of calculating the eigenvalues of a real

symmetric kernel no error estimate seems to be known for

the oldest and most natural method which consists in performing the integration approximately by a formula of numerical quadrature,, It is the purpose of this paper to fill

this gap.

To state our problem precisely we have to describe the

method under discussion in some detail. Let the integral

equation be written in the form

(1) f K(x,l;)y(?)d ?

=? Ky(x) * Jo We assume the given kernel K(x,£) to be real symmetric, or

at least hermitian, and squa re-integrablei

41

(2) Kfx,^) = K(^,x), ]|Ik(x,^)| 2 dx d^ < oo . 0 o We wish to calculate the eigenvalues of K(x,^), i<,e, the constants K / 0 for which (1 ) has a solution y(x) such that

(3) 0 < {

I y(?)l

2

d£j < <d , o It is known that the eigenvalues are a real bounded set which

is either finite or has 0 as its only point of accumulation,

even if multiplicities are counted.

The first step in the numerical procedure which we want

to discuss consists in selecting some rule S for numerical
-

3 -

quadrature. We choose a natural number n, a set of n numbers

<...<£j contained in the interval 0 < X < 1, a set of n positive numbers p^ with sum 1 , and form the sum

st = iz p fa

)

\> = 1 *

as an approximation to the integral of the function fCx);

we write

( 5 ) Sf

.

In the second step of the procedure the selected n-pointrule S is used to replace the integral equation (1 ) by a system of n linear algebraic equations (7) in the following

way.

Suppose K is an eigenvalue of (1 ) and y(x) a corresponding

eigenfunction. If we specify x = ?«,( 1,.„.,n) and approximate the integral by S we find from (1)

(6)

3~~ K(^ ,1 )p y(£ )^Ky(^,) (f*= 1,o. .,n).

V =1 C* v v 9 ^

x

S Introducing n unknown constants

y^

and a new eigenvalue K

we form the system of equations corresponding to (6)s

n

( 7 ) 31 K(~ )p y^ = fc y (M= 1 ,2, . . . , n)

V =1 v V > f* \

In the third step the n eigenvalues k of this algebraic

problem are computed by any of the several available methods.
-4-

In the last step we assign these n values as approximations

to certain of the(possibly infinitely many) eigenvalues

of (1)o There is a natural way of doing this« We remark

S S that every K is real; for k is an eigenvalue of the n matrix

(8) rS = X X

which is similar to the hermitian matrix (p 2K( T , ? )p 2

) 0 h v v s Since each ^ as well as each k is real, and since the ^

! s have no other limit point than 0, it is natural to take the

S largest positive k as an approximation to the largest positive K , the negative largest in modulus as an approximation to the negative K largest in modulus, and so on.

Whenever a term we need is missing we shall take 0 in its place

(following WEYL [11]? p. *+^3)

^ since almost all of the s are near to 0. To formalize this correspondence we define

K

p (p = 1,2, oeo ) to be the p-th positive eigenvalue of K

(counting from the top and taking regard of the multiplicities) provided a p-th positive eigenvalue exists; if not,

S S

we define * = 0, In the same way we define K_ n (p=1,2, ooa ) P P

5 to be the p-th negative eigenvalue of K (counting from the

bottom) or 0; and similarly we denote by k^(^=+ 1,+2,*..)

the eigenvalues of K? possibly completed by zeros. The

notation is such that
- 5-

(9) K

1 > , K

_! < *_2 <.. o“^0

,

K

1 S

S

K

2

S K

_-j

S

< 12s <...^o .

(In what follows we shall refer to these numbers as the eigeng values of K(x,2f) and FC , though this is not the usual deg finition as far as 0 is concerned.) We take k" as an

approximation to for every integer

^ ^ 0, and ask for a method to calculate error bounds in the case of any given

hermitian kernel K(x,0 and any given n-point-rule S for

numerical quadrature.

It appears from BlJcKNER's report [1] that so far only

limit relations for n - oo have been obtained. HILBF.RT, in

his first proof for the existence of eigenvalues for a continuous real symmetric kernel [7], proved that for the special

n-point rules S^defined by = V/n, p^ = 1/n we have

as n—>oo. BOCKNER ([1], p. 3&2? [2], p. 110) extended

this result to a more general class of rules. In addition

he obtained, for certain rules T

n

, the stronger result

( 10 ) lim

n —

oo

n

P (K(, ~ ) = 0

under the assumption that K(x,^) has a continuous p-th derivative with respect to x. (see [1], p. 113; [ 2 1

,

p. 370).

His proof of (10) (related to perturbation theory) involves

in the pre-limit stage the unknown eigenfunction y (x) be-

$
6-

longing to „ For this

a suitable way to answer

develop a new method 0

reason it does not seem to provide

our question*, and we prefer to

2, Reduction to an approximation problem for functions of

two variable S o The difficulty lies in the fact that we have to compare kernels and matricess it would be easier to deal with

two kernels? or with two mat rice s„ In fact? there is an

important theorem of WEYL p 0 *+4-5) which allows us to

compare the eigenvalues of two hermitian kernels Kfx,^) and

G(x,£), or of two hermitian matrices K and G of the same

order? without any previous knowledge concerning eigenfunctions

or eigenvec torso To formulate the theorem we define, for

any kernel or square matrix H, the nonnegative number

f11) || H || = max Itj!

where Yj runs over all eigenvalues of H» Then WEYL's theorem

implies that

(12) |k

? -

J?

l < I|k - gII (^ = + 1 , ± 2,.„.)

whenever K and G are two hermitian kernels? or two hermitian

matrices, with eigenvalues and ^

numbered according to (9)»

To overcome the discrepancy between kernels and matrices

we consider those hermitian kernels Kfx,^) which have the

special property, with respect to a fixed rule S for numerical
-7-

auadrature, that the matrix derived from K(x,?) by S' has

the same eigenvalues as K(x,£) has:

( 13 ) kJ* = K^> ( ^

- + 1, + 2 , . . . ) »

These kernels will be said to allow s. Postponing the question of how to construct kernels allowing s, we state

Theorem 1 . Let K(x,Zp be a hermitian kernel and S a

rule (b) for numerical quadrature . Then the eigenvalues

5 and defined by (9) satisfy

M*0 K -K^ S

| 1 || K - g|| + |l K

S - GS

II (q = 1 1 , ± 2,

for every hermitian kernel G(x,S;) Q.lli>wina S.

- S Proof . If the eigenvalues of G(x,£;) and G' are denoted

S .. I . ... S by ^

and ^

then by ( 13 )

S

Hence

V K

?

f = lv&/ +|K

? “ ti !'

Since K(x,^) and G(x,^) are hermitian, and since k and G

a

may be transformed into hermitian matrices simultaneously

by a similarity transformation which does not affect II K^-G^ II we have by (12)

!k

? - jf^l i l| K - G II , Ik* - ^

S

| Sl|K

S

- G

s

||

which proves theorem 1

.

Corollary 1

1 . Let K(x,<5j) be a hermitian kernel , S a rule

(4), G(x,^) a hermitian kernel allowing. S which coincides

with K(x,£) on the mesh points determined by S, that is
- 8-

n 5) G( ^r

? V

" = 1 9 2 ^

o o o ,n)

Then

0 6 ) I K

? - K

?

S

| s 0 K - G |) <? = ± 1 a J~+ 2C- Q) a o )

In order to facilitate the numerical application of these

theorems we recall some simple estimates for the right hand

sides* For arbitrary (not necessarily hermitian) kernels

2) (or matrices) H we have

1 n (17) l| H || < sup /|H(x,?)|d?

” X 0

l| h || 2 /J|h(x, 5)|

2 dx d<f (or < ZT|h(Xv | 2

) go

”

or < sup | h i) - f*. V=1 P

(18)

(19)

r

II H II < sup | H(x,?) I ~ x,?

(20) If |h

1 (x,f) | £

H

2 (x,f) then I|h

1

|J $

If K(x ? f) - G(x 9 ^) < M then by (19) and (17)

n

ll K - G II < M c II

S - G

S

i| < M X p . = Me “ - V = 1 v Hence we have the convenient

Corollary 1 n 0 It_ K(x ? $-) and G(x 9 <^) are hermitian kernels

such that G(x allows S then

2) (17) and 08) are easily derived from

For (20) cf o JENTZSCH £.9] and FROBFNTUS [ 7 ], p

'

0 51 6 ,
( 21 )

-9-

This leads us to consider the following question: How well

can a given kernel K(x,<f) be approximated by herrnitian kernels

allowing a given rule S?

3 • Construction of kernels which allow S

„

Let us say that a set of functions g.j (x) ,

. . . , g (x) admits

a given quadrature formula S if each g^(x) is squa re-integrable

over 0 ^

x ^

1 and the scalar product of every two of them

can be calculated exactly by S, that is

Every set with this property provides a method for constructing kernels which allow S in the sense of ( 1

3

) s

Thftnpftm P. I. fit n . f v ^ n ( x 1 adni t anrl let. c. be

( 22 )

m m

(23)

— i j — >

is a hermitian kernel allowiaq S . Proof > Let us denote the expression (23) by K(x,£).

Evidently K(x,^) fulfills (2). Hence it will be sufficient

to prove that the eigenvalues k i- 0 of K(x,^) coincide with

s s the eigenvalues k / 0 of the matrix IT defined by (8).
- 10 -

Let y(x ) be a solution of (1) and (3)« Since K i- 0 we

find f ron (1) and (3) that y(x) has the form

(24)

m

y(x) = c g (x)

* =1

with certain constants c 0 Now (22) shows that (6) holds

not only approximately but exactly. This means that

(25) K =K? y = y(Z ,) ( V = 1 , 2, 0 0 0 , n)

>) V satisfy (7), Moreover we know that some y^/ 0 since by

(24) and (22)

0 < / ly(f) I 2 d? = S | y | 2 pj yJ

2

- V =1

Hence K is an eigenvalue of K , and the linear mapping (25)

of eigenfunctions into eigenvectors preserves linear independence. As a consequence the multiplicity of K as an

eigenvalue for K is not less than for K(x,^) 0 So we know

2- (K^ )^ > X To prove theorem 2 it is sufficient

to show that here equality holds. We have (from SMITHIF.S

[ 1 0 1 formula 5o 1 « 1

)

11

<ET Kq - //k(x,^) K(% ,x)dx df

which can be written, by (22) and (23), in the form

n n t

K

f

£ S “VV,«VVv

= trace (KS

) 2 = 21 ( K

S

) 2

. ? f
- 11 -

,s

This completes the proof.

Rema rks . (a) The eigenvectors of K° are explicitly

given by (25) in terms of the eigenfunctions of K(x ? ^)„

(b) $he restriction of theorem 2 to hermitian kernels can

be avoided by a slight modification of the proof (calculate

trace (K ) p

$ p = 2,3>...).

Theorem 2 suggests the following procedure to obtain

Q

bounds for Ik^ ~ I s (I) Select a set g^ (x) , . . . , g^(x) which

admits S. (II) Select constants c^ which make ( 23 ) an

approximation to K(x,^). (Ill) Apply theorem 1 or its corollaries.

Postponing the discussion of I to section 5 where several

rules S will be dealt with we turn to the approximation

problem II.

*+. Reduction to an approximation problem for functions of a single variable . We want to reduce our problem of approximating K(x,^)

by ZZc g^(x)g^(^) to the ’’one-dimensional” problem of

approximating functions f(x) by I c^ g^(x) . For the sake

of simplicity let us assume that

( 26 ) K(x,^) = K(|‘,x)j K(x,^) continuous in 0 < x,| < 1.

Suppose we have defined an approximation operator A which

assigns to every continuous function f(x) a linear combination of m fixed arbitrary functions g^(x)s
- 12-

( 2 ?)

where the coefficients c tJ depend on f » Then it is natural

r

to assign to the kernel K(x?^) the new kernel

( 28 ) G(x,^) = A

x A^ K(x?f)

where the subscripts denote the variable to which the operator is to be applied? and the conjugate complex operator A

is defined by

Under suitable assumptions every error estimate for the

approximation of f by Af yields an error estimate for the

approximation of K(x,f) by G(x,^)s

Theorem 3 « Let the operator ( 27 ) be linear^ i,e, , for any

3 ) It will be seen later that this assumption is a weak

point in our method of reducing the two-dimensional to

the one-dimensional approximation problem,, It would

seem desirable to have a reduction method avoiding

the constant c„

( 29 ) Af (x) = Af (x) «,
- 13-

for every function f(x) continuous in 0 < x < 1. Let K(x,^)

satisfy ( 26 ). Then ( 28 ) defines a hermitian kernel G(x,?)

of the form ( 23 ) such that for 0 < x , £

< 1

(31 ) | K(x,f) - G(x,<f ) | 1 (1 + c)sup | K(x,§) - A K(x,2p |

.

x ? Jf Proof . Since K(x,^) is continuous we can approximate

K(x,|;) uniformly by a polynomial;

r

^

<5? ^=0 <r^t> V

where £ > 0 may be prescribed arbitrarily small; and by symmet-

(32) |K(*,?) - ± ^ x'f I < f

rizing, i.e. exchanging x for^, we can achieve a ^

= a^ since

(S' " > K(x,^) is hermitian. Define h^(^) - £ . Then by (30)

(33) ArK(x,^) - ST ST a ocr S ^=0 <s-=0 ^ ^ (<f) | £ c£ (0 < x,^< 1 )

which shows that A^K(x,?) can be approximated with arbitrarily

high accuracy by a polynomial in x, for every fixed 2; • Hence

A^K(x,£) is continuous in x, and we may apply A^ so that ( 28 ) really defines a function G(x,?). Applying A to

A

(33) we find

(3“t) |G(x,f) -

^ =0 <0 =0 ^

a h (x) h (?) | < c 2

£ (0<x, ^

< 1)

Since a = a , the double sum is hermitian. Hence by letting

£ ->0 we find that G(x,?3 is hermitian. In addition, since

the h^(x) are linear combinations of g-j (x) , . . .

,

g m

(x) we find

that G(x,?) has the form (23) » Finally
-14-

| K(x,f)-G(x,^) | £ |k(x,?) - A

x K(x,f)| + J A

x [K(x,f)-A K(x,f) ]|

< M + c M

if

sup | K(x,f) - A K(xi)| = M,

x, ^

X

since

|k(x,?) - A^K(x»?)| = | K(x,f) - A^K(x,f)j

= |K(f,x) - A K(x,f)| =|K(f,x) - A^K (f,x)| < M.

Remarks » (a) From (3*0 we see that also A- A K(x,^)=G(x,|-)

s

x

(b) If A is an integral operator with a real non-negative kernel

such that for f

Q (x) = 1 we have AfQ (x) = 1 then (30) is true

with

( 35 ) C = 1 o

To prove this let f(x) = u(x) + iv(x) 0 Then we have, with

M = sup | f (X) | ,

Re[A f (x) ] = A u(x) = A[M f (x)

- A[M f

Q (x) - u ( x ) ]

Since M f

Q (x) - u(x) > 0 a nd A has a positive kernel we conclude that

Re[ A f ( x ) ] < A[M f (x)] = M„

In the same manner we find for every complex number ^ with

modulus 1
- 15-

Re ^ [A f (x) ] = Fie LA ^

f (x) ] < M 0

Hence j A f(x)| < M» This means that (35) holds.

5 o Five special quadrature formulas .

A, We begin with the simple rule for which the

abscissas £; v are the left end points of n subintervals of

equal length and all weights p^ are equal. We denote this

rule by Leq

n

s

(36) Leq

n

s ?

v = ^ =

n (V = 1,...,n).

Obviously the n piecewise constant functions

(37) g.(x)

1 if V" =

x < n = 1...

1 if V = n and x = 1

0 else

. ? n)

admit Leq

n in the sense of (22). (In a similar way, for

every n-point-rule S a system of n piecewise constant functions admitting S can be constructed; see [1], p. 109).

We form the kernel

G(x, f) = y~ K ( ? 5 J )g (x) g (?) Z— iA M, V

f*=1 V=1 r r

which is constant in every subsquare

< x <P , ^ < I < i

n = n

5 n =. S n

(|U.

, V 1, 0 .. ,n)

G(x,f) is of type (23) and coincides with K(x,^) at the mesh
- 16-

points ( ^5 ?y) * Hence (16) holds. To estimate )| K — G|| we

assume that K(x,£) satisfies a Lipsehitz condition with a 2

known constant L at each of the n mesh points?

(38)

|K(x,?) ~ K (f*n>i i

L( f x “ |! + ^

‘

This condition is certainly satisfied if K(x,p has a

partial derivative K (x,£) such that |K (x,f)|^ L for every

x, 5 ® (38) means that

|k(x, 5) - G(x,?) I 1 L P(x,f)

where

(39) P(x,J>) = p(x) + p(?) ; p(x) = x - £ if

f < x < f

J

1 . By (20) we havejK - G|| < L IIP II . By (39)

every eigenfunction of P(x,^) has the form c^p(x) + C 2

? hence the eigenvalues of P(x,j;) can be calculated by solving

the quadratic equation

4 1 4

^

2 - 2v J pd* + [ / pd* j

2 - /

p

2 00 dx = 0.

0

The roots are

Hence II P II = ^

by definition (11). We have proved

Theorem *+. Let S = Leq^ . Let K(x ,

)

be a hermitian

kernel satisfying the Lipsehitz condition (38). Then
-17-

(^1

)

I K

^ - K

|

I 4 - 1

> ± 2,...)

where

C = |

+ sP^ <

1 . 08 .

We remark that the constant C cannot be replaced by any

smaller constant independent of K(x,f ) « For in the special

case K(x,^) = LP(x,f) equality holds in (41 )fox» ^

= 1 *

Bo By the same method we can treat the rule Meq whose

n abscissas are the mid-points of n subintervals of equal length;

(42) Meq

n

i S

y

= J P„ = ±

=• 0 < » < ») .

Changing (38) and (39) in an obvious way (with p(x)= min |x-^|)

(X p

we arrive at

Theorem 5. Let S - Meq

n

» Let the hermitian kernel

K(x,2f) satisfy

|K(x,5) - K (^J. , 2V_=1|

< L (|x - + U - 2^!|)

(43)

if |x - 2/i^i| <; L.

, |?

2 V -1

f . 1_

n I 2n

Then

(44) I K - K

S

j < ~ I 0 ?1 = n ? ^ ' = (^ - ± 1, ± 2,...)

where C = ^

+ -s/j^ < . 54 is the best possible constant .

Co For the trapezoidal rule
- 18-

(45) Tra C _ V — 1 _ _ 1 \r n-1 > p 1 " pn “ 2n’ p 2

•= pn-1 =... 1

(1 < v> < n)

find similarly, replacing (37) by step functions which

i jump at (V- yp/tn-l) instead of 7/n;

Theorem 6. Let S = Tra „ Let the hermitian kernel K(x,£)

n

» satisfy

|k(x, 5) - K(£, *)| < L(|x - £|+|?- *|)

U l X ~ n| < k> |?“ n | < k •

Then

( 46

)

|

I ~ n-T ^?

= — ^ 5 — 2,o.o)

1 FC where C = y

+ Vyy < .54 is the best possible constant .

D. We now turn to Simpson’s rule Sim

n defined by

(47) Sim s ^ = (V-1)h, h = ~T (V = 1,2,„..,n; n = 2m+1 ) n y n i

Pi = Pn=3> P3=P5=..-=Pn-2 =

I*

1

, p 2=pl+=..,=p

n

_ 1 = ^ .

In order to get good approximations we try to avoid step functions. Since Simpson's rule integrates exactly every continuous function f(x) which is quadratic in each of the

m = ( n-1 )/2 intervals

(48) £, < x < £ ({*• = 1,2,...,m),

2«-1 " 2jw + 1
19-

eaeh set of continuous functions which arc lineain each of

the ra intervals (*+8) will admit Sim

n

» We choose any set of

m+1 linearly independent functions g.j (x) , » , , , g^-j (x) of this

type. Their linear combinations exhaust the continuous piecewise linear functions with vertices at ^

n

.

In order to apply theorem 3 we define an approximation

operator A by interpolation at the m+1 points °

W) Af(x)

= Y V

(X^’ Af(?

2f<-1

) = f(

?2ft-1

) ( (* = 5 e o o 5 m+1 ) o

Obviously A is linear. The best constant c in (30) is C =1 f since the piecewise linear function Af(x) attains its maximum

1

, and there coincides with f(x). Let f(x) have

a continuous second derivative in 0 < x < 1 , and

(50) | f"(x)| < M,

To estimate the error |f(x) - Af(x)|, it will be sufficient to

consider the interval ^

< x < that is 0 < x < 2h , In

addition, since neither our assumption (50) nor our conclusion

(55) is affected by subtracting a linear function from f(x),

we may assume without loss of generality that

(5D f(0) = f(2h) = 0,

ih

f(x) = /

L(x,?) f"(f)df (0 < x < 2h

)

Then

(52)
-20

where

(53)

(*,$) =

- ^ (2h-x) if 0 ^ x ,

(2h-?) if x S ? < 2h

,

x

2h

Hence for 0 £ x *< 2h

2^} (54)

J f(x)|< M

J | L(x,?)| d£ = M ^§^1 <«li

From (51) we have Af(x) = 0 for 0 < x | 2h, hence (54) gives

(?5) |f(x) -Af(x)|^ (0 < x < 2h).

Since the intervals [ ?j',' fj], . . . may be treated in the same

manne r

g (55) holds for 0^ x=< 1 . Now let

«6 ) 1ST) - K(?

,x), |K

xx

(x, ?)|<m

where the derivative K

xx

theorem 3 states

is supposed t0 be continuous>

^

that G(x,f) = Aj£ ^ K(x, ? ) is a hermitian

kernel allowing Sim

n and approximoting K(x,?), on account

of 0=1 and (55), with a uniform error < Mh 2

. Hence we have

by (20)

(57) for S = Sim

n

J S 2Mh 2 - —

(n-1 ) 2 < ? = ±1 . ±2 ) o •

If we had been able to make coincide G(x,f) with K(x,f) at

all mesh points then the constant 2 in (57) could have been

replaced by 1. But obviously our m+1 = ( n+ 1)/2 functions

9(i(x) are too few to do this; we would need n functions g (x)

which are linearly independent on the n points F So we^are

led to search for m more functions Va(x) , . . .

,

g'(x) which
-21 -

coraplete the m+1 functions g (x) defined above [after (48)]

to a system g-j (x) , . • • , g Q (x) still admitting Sim

n

. It may

be checked by a simple calculation that the step functions

M if lx - E- I < Q

9ra+1+^ x ^ -}0 else ^ ^ (“ =

4) do us this favor . Using these functions we may proceed

as before, replacing (49) by

n Af(x) = V” a g (x), Af(<f ) = f(5) ( V = 1,...,n). Z— v 9 9 v 9 = 1 We have, say in 0 < x < 2h,

(59) Af (x) =- l2L~ f(0) + (1 - 2^)f (2fa ) if

J x-h | > 2h

f(h) + (x_h) if | x-h | < y

1

.

Hence |Af(x)| < J

supjf(x)|, that is (30) holds with

(60) c = *

3

.

To es tima te J f (x)-Af (x)| we assume,

Then by (59)

( f (x)

f(x) - Af (x) = <1 f ( x ) - f(h)

as before, (50) and (51).

if (x-h) > ,

if (x-h) < ^

.

4) The author is not aware of any convenient system of continuous functions 9nr)-2 ?, * , ’^n serv^ n 9 our purpose. This lack

of information concerning (22) seems to indicate that an

investigation of numerical quadrature of products of two

functions is desirable. Such a theory would be useful in

numerical applications of Hilbert space.
22-

Kence by ( 5*0 > still in 0 < x < 2h,

(61) j f ( x ) - Af (x) | < M < M if

| x-h / > y

1

,

and 2h

| f (x) - Af (x) J i M / j L(x,?) - L(h,^)| d? if | x-h

j < y

1

£3 where L(x,^) is given by (53)® The right hand side is found,

after some computation, to be

( 62 ) M

(ll%^p-) < M if l x_h ( S f

1

•

From (61 ) and (62), we have, for every x with 0 x 1 2h, and

hence in the entire interval 0 < x ^

1

2

( 63 ) | f (x) - Af (x) j < M :jy~ (h = yy)

instead of (55)® Using (16), (19), (3^ ) end (60) we find

Theorem 7® Let S = Sim . If K(x,F) is hermitian and has

a continuous second derivative such that I ^

xx (x,f) | < M then

C64) IVK

r

1 =

CM

( n-1

)

2

where C = (1 + -|) < .75®

It should be possible to replace C by a substantially

smaller constant by using ( 17 ) instead of ( 19 ).

E« Our last example is Gauss' rule Gau

n# Its (irrational)

abscissas and (positive) weights are uniquely determined by

the property that
-23-

A

f V T

(65) Gau

n

s lx dx - Gau^x ( T* - 0, 1

, , , ,

, 2n-1 )

o

n_'j From (65) we see that the system 1, x, 0 o<,,x n admits Gau

n

„ Hence we find from theorems 1 ,

9 3

Theorem 8, Let S = Gau . Let there exist a linear _ n operator A which assigns to every continuous function f(x)

(0 < x < 1) a polynomial A^f (x ) of degree < n such that

(66)

| AR

f (x)

j < c n max

| f (2j) | (0 < x <. 1 ) , and for some p<n

( 6

) j f (x) - An f(x)| < c< p) max

j f ^(f) J ii f

^

is continuous ,

/ \ with constants c n and c^ p ^ independent of f. Then we have for

every hermitian kernel K(x,^) which has a continuous p-th

partial derivative K^ p ^(x,^) with respect to x,

(68) J k __ k

S

I < 2(1+c )c^max ) (x, £) )

' f f ' x,f '

From the extensive literature on approximation, several

operators A

q

are either immediately available or can be obtained from trigonometric approximation operators by substituting X = 4 (1 + cos

<Jj)#

COO [

“!• j

. Thus, the ’’trigonometric” operator defined by JACKSON

5)

( [ 8 J p, 3) leads to the ’’polynomial” constants'^

5) These constants are better than those given by JACKSON,

p, 17. They have been calculated by Dr 0 P. HENRICI

He found that if ) f C p ) (x)| £ 1, f (1> ) = . <, ,=f ( P- ^ ) (‘i‘)=0

and f (£ + i cos 0) = g(0O then |g'P^(0) J ^ H p where H< .64

for 1 < p < 6, It is known [4] that H < e/2 for 1^p<oo.
-2b-

(p < 6) ,

(p > 7) .

Since JACKSON’s operator is an integral operator with a nonnegative kernel, (66) holds with c =1 . Hence we have

Corollary 8’. Let S = Ga

u ^ . Let the hermitian kernel

K(x,^) possess a continuous p-th derivative with

respect to x „ Then we have for n _> p

P

(70) Ik, -4 I £ ^ (rir) max |k(p) (x,?)|

) > ~ n 1 X'Z where d = 1 ,92„ if p £ 6, and d = 4.1 if p > 7.

FAVARD [5] has defined a trigonometric operator which

leads to the ’’polynomial” constants

(71) c (p) =

n

( .*62) p

(1.36)

P

n

'

(2 < p < 6)

,

(p > 7).

These constants are better than those given by (69).

Unfortunately no constant c n satisfying (66) seems to be known

for FAVARD's operator, hence we cannot use (71) to improve

(70). However, there is a special class of kernels where we

can do without c , viz. kernels of the form n

7

( 72 ) K(x,£) = h

1 (x+f) + h 2 (x-p (0 < x, £ < 1).

We prove

Theorem 9- Let S = Gau^, and let K(x,^) be a hermitian
-25-

kernel of type (72) » If there exist compl-ex polynomials

p.j (x) and p2 (x) of degrees < n such that for some constants

£l ’ £ 2

(73) | h 2 (x)-p2

(x)| £ £2

(—

<x<1 )

, J h-j (x)— p-j (x) J <_ £ ^ (0<x<2)

then we have

(7^)

|

I - 2(8.^ + £ 2

) (^

= ±l?±2,. 0<>)o

Proof o Put

M(x,?) = p-| (x+^) + p 2(x-?’) „

Then we have from (73)

| K(x,^) - M(x,^)| <£^ + £

2 (0 < x, f

< 1)o

Since K(x,^) is hermitian the kernel

G(x,?) - £ [M(x,f) + M (?,x)]

also satisfies

) K(x

)

- G(x ? ^) | < £

1 + £

2

o

Moreover, G(x,{f) is hermitian and a polynomial of degree < n

in x (and in £ ) ; hence G(x,^) allays Gau^ by theorem 2.

Theorem 9 now follows from corollary 1 M

. We may apply FAVARD’s operator to h^ (x) in 0 < x < 2,

and to h

2 (x) in -1 < x < 1 „ Since we now have intervals of

length 2 the numerators occurring in ( 71 ) have to be multiplied

by 2o We obtain
-26-

Corollary 9'. Let S = Ga

u

^ . Let K(x,lp be a hermitian kernel

of type (7 2) such that and h^have continuous p-th deriva -

tiveso Then we have for n 2 p

(75) | K - kf | < 2.(^)

P

max fh^tj + h^P^u)]

? 1 0|t<2

-1<u<1

where d = 1 .3*+ if 2 < p < 6, and d - 2,72 if p > 7 .

F *' Once more Leq , Whenever K ( x , £ ) is periodic then

better results may be expected from the simple rule Leq^

than from Gau . To show this we note that the system of n

n exponential functions

(76) - - -

% 2,ri jir-i (

IB) • » • ) OS 1 g (x) = e^ where u = < 1 -m, . . . ,ra if n = 2m+

if n = 2tn+2

admits Leq . (It also admits Meq and Tra which, on account

n n n

7

of the periodicity, is not essentially different from Leq^.

)

Repeating the reasoning of section 5E, we arrive at the

following analogues of theorems 8 , etc.?

Theorem 10. Let S = Leq^. Let there exist a linear operator

which assigns to every continuous function f(x) of period 1

a linear combination of the functions g^(x) defined bv ( 76 ) such

that for some constant c

(77) I A f(x)| < c max f(?)j (~oo < x < oo

)

in se n ^

and further, if f^^(x) exists and is continuous^ for some
- 27-

(p) constant c_ — K

n

( 78 ) • |f(x) - A

n f(x)J < c n

( P } max | f" P ^(^)|, (-oo< x < 00 )

„

If K(x,$-) is a heriaitian kernel of period 1 and K^(x,J)

= (3/dx) P K(x,f) is continuous (in the entire ( x , ^

)

-plane) then

(79) | K

e -^S

| < 2 ( 1+c

n^

c n^ max | K

(p) (x ? ?)|.

’ x,^

JACKSON [8] and FAVARD [3] have given trigonometric operators

fi 1 A

r leading to the constants '

(80) c =1 ; c (p) =

n ’ n

(Ja56) P

^ n-2 ; (p > 1)

(81) c (P) =

n n (p > 2)

Hence we have

Corollary 10’. Let S = Leq

n

« Let K(x,3p be a hermitian

kernel with period 1 . Then

( 82 ) K - K?

| < 4 (

g ^2> max I (x ? ^) | • ? 5 “ n ^ x,f

To theorem 9 there corresponds

Theorem 1 1 . Let S - Leq . Let

n

6) JACKSON [6] p. 10 gives the constant (3/ra) P for approximation by trigonoaet rieal polynomials of order ^ m in 0^x^2tt.

In our case we have m= ( n-1 )/2 or m=(n-2)/2, and 3 is to be

replaced by 3/2*< .48 for the interval O^x^l . This leads

to (80). Similarly (81 ) is obtained from FAVAFiD 1 s trigonometric constants.
-28-

K(x ? £) = h

1 (x+?) + h 2 (x-^)

be a hermitian kernel* with function s h^ (x) , h 2

(x) of period 1 0

If there exist linear combinations e^(x), e 2 (x) of the exponentials functions (?6) such that for -oo < x < oo

( 83 ) |h (x) - e (x)| < £ («= 1,2; £ constant )

Oi TM

then we have

(8^)

| K

f — k| I i 2(B^ + £2

) (f= ± 1, i 2,...).,

Here the approximation operators work even better than in the

polynomial case since, on account of the periodicity, the duplication of the interval does no ha rm« Using FAVARD * s constants

(81 ) we obtain

Corollary 11', Let S = Leq

,

Let K ( x , fp be a hermitian

kernel of type (72) where h^(x), h

2 (x) have the period 1 and

a continuous p-th derivative such that

j (x) | < (-00 < x < 00 ; W = 1

, 2 )

Then we have for

^ = + 1 , + 2 , • • •

(85)

I K

? -k|| < 2 (-^5)

P

(^ + l

2

).

Remark , Under the assumption that K(x,lp is analytic in

a known neighborhood of0£x<1,0££*<1, especially

convenient bounds.of {k^, - for an arbitrary quadrature

formula S, might be obtained from the error estimates given

by DAVIS for the approximation of analytic functions [3],
-29-

6. Two Numerical Example s,

I, Suppose we want to calculate the eigenvalues of

K(x,f) = ex? (0 $ x,f < 1)

using the trapezoidal n-point-rule . How large should we

choose n to be sure that the error of each eigenvalue is

less than .01? This question is answered by theorem 6.

Using the trivial estimate Ik (x,?)| < e we find

I u — kN* ^ ^

( p = + 'l +2 )

Hence n > 148 is sufficient. If we want to answer the same

question for Gauss* rule we may either estimate

and apply Corollary 8* for some suitable chosen value of

p, or we may search for a good explicit polynomial approximation to K(x,^) and then apply theorem 2 and corollary 1".

The first method gives the estimates

P

I be( ^ °

• Vi ' K -K

? 1 4e(fH-)

< 4.1 s p

n-1

(p < 6, p < n)

(7 < P < n)

The smallest bound results from choosing p = 6; this leads

to n 2 8. In the second method it is natural to use the

Taylor expansion with center If 0 ^

z ^

1 then

l *-i i V

-

( . i f j;-) n

ez=eiez-i=ei 21 (z4) + r , In

y-0 V I a’ I n | — n
-30-

Putting

i n-1 ( x. iO

GCx.,5) = :>

we find from ( 21 ) the estimate

feiul

^

K ii if e

This shows that already n 2 5 is sufficient forjK^-K^ M j< .01.

II. Sometimes our theorems can be applied in the inverse

direction; We have some information concerning the eigenvalues of K(x ? ^) and ask for similar information concerning

S the eigenvalues of K . For instance, let

( 80 ) T^lT cos 2?r(x+^) (0 < 1 )

Since K(x,^) > 0 and /K(x,^)d^ - 1 for every x, we know

that K-, = 1 , | | < 1 (^1) (cf. JENTZ5CH [ 7 ], FROBENIUS [*+])

Suppose we want to perform some iteration process with the

matrix K which is derived from K(x,^) by one of the n-point

quadrature formulas S discussed in section 5? knowing that

the process will converge satisfactorily if all | I < 1 . 1 . 7) [low large should we choose n in order to secure convergence?

The kernel (80) is pretty bad for numerical integration

7) This question was proposed to the author by Dr. J. Todd

and gave rise to the present investigation. The kernel

(86) appeared in a numerical procedure for conformal mapping of the interior of an ellipse with axis ratio 5 onto

the unit circle.
-31-

since it has a high crest along x + = 1 „ The derivatives

have the upper bounds

( 87 ) P = 0 1 2 3 ^ 5 6

I 5 49.7 1270 85800 683x10 444x10° 492x10°

Using these values (and p = 6 where K^^(x,<f) occurs)

we find that

( 88 )

(

(89)

and hence |K^

(*ti ) (mo

Leq

537 Meq 269

I 4

1 . 1 )

(64)

Sim

1 35

K

? ~K^|< .1 - ± 1, ± 2,

if S is one of the following rules?

(70) (75) (82) (85)

Gau2l6 Gau 134- Leq 110 Leq 36 '

0 9 0 )

Comparing the results for Simpson's and Gauss' rule we see

that our Gauss estimates (70) and (75) are far from being

good (they could be greatly improved if the constant c n for FAVARD's operator turned out to be near to 1), In (75)

and (85) the fact is used, that K(x,|j) is a function of

a single variable. (82) and ( 85 ) use the periodicity of

K(x,Sj) . Better results than by our general theorems, are, of

course, obtained if the special structure of our kernel (86)

is used. To do this we develop K(x,^) into its Fourier

series?

K(x,?) = y~ (|)

,J|

e

27ri j (x+ ?\

j= -00
-32-

and then, say for n = 2m+1 , choose

m — 2sl j l 27rij(x+^) G(x = ? 5 } = { V

j=-m

\

~ c g (x) g (f) ^1— Of (3 A

i=-m '

'

with c _

2nIW = S. _ (-f) and g<X)defined by (8l), We have

P

%~[b J Of

|k(x,^) - G(x,?)| 1 2

m+1

n+1

(?)

J = 6(ff

‘ ’= 6(|) 2

3' ‘3’ '3' j= null e n c e by ($+)

) K p - K0 12.(|)

n+1

2n 2

From this estimate it follows that already Leq

2 ^ is sufficient.

Which is the smallest value of n such that S = Leq satisfies

n

( 88 )?

In the case of the kernel (86) this value of n can actually

be determined. For the eigenvalues of K(x,^) can be calculated

exactly, as it is always the case when K(x,If) = f(x+ip

+ g(x-Jp with periodic functions f,g (the eigenfunctions have

the form ae In a similar way the eigen- 2lrirx + be~2,rirx

) Leq

n values of K can be calculated. The result is (after

a convenient change in the numbering of the eigenvalues):

( q

^ ( 0 = 0 ,

1

, 2 ,...) , p

(90) K =i \e\ (q = j)

f 1 -q ^ (^=-1,-2,...), 2

t 9 1+q n 1-q n

(9D i+jaJT

2 ^

(^- 0,1,2,...,[2])^

n-1

( 9

=

else «
-33-

F rom this it follows that for n < 10

I K-r- ^’"1= Ik_J = (#)

? > .1,

Leo

| but for n J> 11 always ' #'j < . 1 . Hence n = 11 is the

lowest admissible value.

Perhaps it will be felt that all of the results mentioned

in the table (39) are not very satisfactory. However, one

must bear in mind that no general approximation theorem

which is based on uniform bounds for some derivative is likely

to give good results for the special kernel (86) since this

kernel has high values of K^^(x,2f) only at a small part of

the unit square, though our estimates are valid for kernels

which have everywhere derivatives of the order of magnitude

allowed by ( 8

). Better results for the kernel ( 86 ) could

be derived from approximation theorems involving say

max / U

(p) (x,?) I dx instead of max | (x , ^) |

. A clo se look at example II will show that there are at least two

more directions where we may look for refinements.

From (90) and ( 91 ) we see that, for fixed n, the errors

depend on ^

in such a way that they are small for

small !^>| (they attain their maximum value for

q

gg ± n/2).

This means that our estimates which are uniform in ^ are

unnecessarily bad if we are interested, as usual, only in

the first eigenvalues (those of large modulus). It would
-

3

^-

be worth searching for estimates of the type

(92)

'? where depends on K(x,<p, S and ^

, and is small for small

Another direction for improvements is suggested by the

fact that, in our example, we are interested only in onec sided bounds (namely upper bounds for f I ). For some

quadrature formulas S there exist much better upper bounds

for I Kp | , and lower bounds for JkJ, than those implied by

^

j $

' r estimates of f K^—

ou . An example will be given in the

next section.

7. One-sided bounds for Meq^

Theorem 12. Let S = Meq

^ be the quadrature formula

defined by (4-2), Let K(x,^) be a herraitian kernel with a continuous second derivative K (x,if) such that K (x,^) I <

(0 < x,£< 1

)

, Then

(93a) KP'M CL'

o f < 2 (p = 1 ,2 , . . .)

(93b) K r K* J ” n

where C =

-g (3 + s^) < *098 is the best possible constant . The estimate (93) is remarkable for the fact that it is

quadratic in 1/n though the two-sided e st ima te (4-4-) far Meq

R

is only of the first order. Applied to our "bad" kernel

(86) theorem 12 shows that I I < 1*1 for S - Meq) ir>. This
-35-

is a very satisfactory result in view of the fact that neither

the periodicity nor the special form (72) of the kernel has

been used.

Proof of theorem 12 . We define a piecewise continuous

2 herraitian kernel G(x,^) which is constant in each of the n subsqua res

( 9

l0 I h’ < s2n (^AjV — 1 )2j , , » } n) j

where s ^

= (2\J-1)/2n ? such that G(x,^) coincides with K(x,^)

at the mesh points determined by the rule Meq

n

s

( 95 ) G(x,£) = K(? ,5 )

)> if (9*+) holds

By theorem 2 this kernel G(x,^) admits S = Meq , hence its

c

eigenvalues are K© » To prove (93a) let p be a fixed

C

natural number. If KJT = 0 then (93a) is true since K > 0

r P =

by definition. Let K* 0. Then the eigenfunctions z (x)

of G(x,2-) corresponding to the first p eigenvalues

(^= 1

? oo. ? p) are piecewise constant. We determine numbers

a^ such that

( 96 ) z(x) = 21

7T=1

a

7T Z7T^ X ^

is normalized and orthogonal to those eigenfunctions y^ (x)

of K(x,^) which belong to eigenvalues > 0 with 1 ^ ^ < p-1

(Such numbers a

7r exist since the orthogonality requirement

imposes at most p-1 linear homogeneous conditions on the p
-36-

unknowns a „) Put

D(x,^) = G ( x ?^ “ K( x >?)

We prove theorem 12 by estimating the integral

AA

J =

(J z(x) D(xj^) z(<£) dx d£

OO

in two ways. From ( 96 ) we have

^

Q

|j z(x) G(x,^)z(f)dXe|£ *>£, K j 00

' P

and from the orthogonality condition

11

/ /

z ( x ) K(x,^)z(^)dxd£J < K 3 00 F

hence

(97) J > Kp - .

On the other hand we remark that z(x) is constant in each of

the subintervals (>)-1 )/n < x < V/n* Hence J will not be

altered if we replace D(x,2f) by any kernel I)(x,^) which has

the same mean value as D(x,^) in each of the subsquares (91+)»

Let us choose for D(x,F) the kernel which is obtained by

"symmetrizing" D(x,^) in each of the subsquares in the following ways With the notation (9*+) define

Obviously f)(x,£p has the same mean value as D(x,^) in each

subsquare, and in addition we know
-37-

(98a) D<? ,f

v

) = 0, =^(x,^) = 0,

( 98b) | D

xx

(x,?)| £ l' (0 < x, ? <; 1 )

We integrate (98b) twice,) using the initial

the result is

|^(x,f ) | < l'£p(x) + p(^)J, p(x) = min

conditions

(* -?y> 2

2

(98a)

Now we can proceed as in the proof of theorem 4. We find

(99) II S| < l'tt,

where ^

is the largest root of equation (40) formed with

the present definition of p(x) . Solving (40) we find

o

'/r.j = C/n with the value of C given in theorem 12. Now (99)

/ 2 gives J £ CL/n . This inequality together with (97) completes the proof of (93a). Applying (93a) to -K(x,?p instead of K(x,Sj) we obtain (93b).

8. Limitations of the method . Though the method of numerical integration can in

principle be applied to every kernel it will give, as a rule,

good results only if the kernel is smooth at least in the

neighborhood of the mesh points (^,fy). For only these n points are used to calculate the approximations , so

they have to represent the whole of K(x,^). In cases where

K(x,£) is not very smooth better results may be expected
-38-

f rora Kilbert • s second method which approximates, after choosing

n orthonormal functions ^(x) , the eigenvalues of the

integral equation (1) by the eigenvalues of the n x n matrix

whe re

[n] _ K

1 1

1,oso, nj

J/© Cx) K(x,ip *,(?) dx

r

V

o'o

1 *- Error estimates for this method (which may be considered as

an analogue of the well-known Rayleigh-Ritz method for differential equations) are being published elsewhere [12],

We mention the following result of type (92): For p = 1,2,

we have

r 'll

( 100 ) 27Tnj JJ l K(x ’?} l^ dx d ?

n n 0 1 -mu2

].

v=i l
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[1] H« Blickner, Die praktische Behandlunq von Integral - gleichunqen . (Ergebnisse der angewandten

Mathematik vol , 1.) Springer, Berlin, 1952.

[2] H. Biiek ner, Konve rqenzaatersuchunqea bei einem alqebraisehen

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[3] P. Davis, E rrors of numerical approximation for analytic

f unctions . J, rat.Mecb. Anal. 2 (1953). PP»

303-313.

[4] C.’de la Valine Poussin, Lecons sur 1

1 approxima tion des

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[ 5] J. Favard, Sur les meilleures ProWd^s d

1 a pproxirna tion de

certaines classes de fonetions par des polv- nomes t rigoBQiWt rigues . Bull, Sei. math. (2)

vol. 61 (1937), pp. 209-224, 2^3-256.

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Universitat Tubingen, Germany

and

American University, Washington, D. C.

September 30 ? 1953
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