Interpolation and orthonormal systems

INTERPOLATION AND O R T H O N O R M A L SYSTEMS (1)

By

J. L. W a l s h and P h i l i p D a v i s

in Cambridge, Mass., U.S.A.

1. Introduction. Let B be a region of the complex z ( = x + iy)-plane.

By L 2 (B) we shall designate the class of functions which are regular and

single-valued in /3 and are such that

( t ) ] I f H2 ,= fflfj~-dx,gy<~. B

I f the region B is suitably restricted, there exist infinitely many sets of

functions {q~*~(z)}, each function of class Lz( /3) , each set closed and

orthonormal in the sense that

(2) ((P;, (f'~) = ~ij-

(See B e r g m a n [3] where a systematic theory of the class L 2(B) is pre-

sented.) Here ( f , g) denotes generically the inner product (2) ff/gax dy. B

In terms of such a closed orthonormal set, each f ~ L2 (B) possesses

the Fourier development

(3 a) f (z) = ~ , a . q~*, (z), n ~ O

(3b) a,, -~ ( f , ,p~),

which is uniformly and absolutely convergent in each closed subregion

B ' C B . The above series need not be defined unless f E L 2 ( B ) , and if

defined, need not converge. It is the object of the present paper to show

how it is possible in certain cases to construct closed orthonormal systems

{cO*(z)} such that a series (3a) may be defined for each function f of a

c l a s s S o f f u n c t i o n s w h i c h is l a r g e r t h a n L2(B); the series

1. Research paper done at Harvard University, partially under Navy Contract N5ori-07634.

2. It should be noted that much of the present paper is valid when the inner product (f,g)t, = fofg ds is used, where the boundary b of B is assumed rectifiable.

J. L. WALSH and PHILIP DAVIS

will represent f in some subregion of B dependent upon f itself. The

coefficients a . need not be defined by (3b), but will be defined by a process

of interpolation

(4) a. = L" ( f ) ,

which, for f EL2(B), becomes equivalent to (3b). That such a process

may exist is suggested by the elementary case of the circle where there is

identity between the Taylor expansion found by interpolation in the origin

and the expansion in orthogonal polynomials over the region I z I< 1.

The method which will be employed to construct such sets is as follows.

We shall require that the functions q~ (z) be not only closed and orthogonal

in the sense of the inner product (2), but be simultaneously biorthogonal

to a set of bounded linear functionals {L*},

(5) L" - - 8.n.

The functionals L* are certain appropriate linear combinations of a preas-

signed set of linear functionals Ln, L*, = ~ ankLk, while the functions k-~0

W* (z) are the solutions of certain minimum problems for the region B as

explained in detail in w It is to be understood here that t h e p r e a s -

s i g n e d s e t o f f u n c t i o n a l s Ln h a v e d e f i n i t i o n s w h i c h a p p l y

t o a l l f u n c t i o n s f o f t h e c l a s s S l a r g e r t h a n L2(B), while in

the case when f ELZ(B), they have equivalent inner product definitions

as well. The sets q~*(z) satisfying these orthogonality and biorthogonality

conditions may be regarded as a generalization of the so-called doubly

orthogonal functions introduced by B e r g m an [1 , 3].

When the preassigned set of functionals {Ln} is "sufficiently close"

to the set

(6) ( f ) = (0) , (n = 0, 1, . . . )

then we prove that the corresponding set {q)*(z)} will have an expandibility

property for some larger class S. ~3) In w 2, we shall deal with the ortho-

gonal functions which arise from the selection

3. Expansions of arbitrary analytic functions in sequences of analytic functions which satisfy given interpolation conditions and are of minimum norm i n t h e s e n s e o f T c h e b y c h e f f have been previously studied [Walsh 15]. These expansions are

INTERPOLATION AND ORTHONORMAL SYSTEMS

(7) L . ( f ) = f ( z . ) , lim z.-----0, z , , ~ B . n-~oo

For zn-----0, these conditions coincide with (6), a case whose related ortho-

gonal functions have been discussed by B e r g m a n , and previously by

W a l s h [12], and W i d d e r [17] for a simply connected region. We shall

prove that if f is regular in a region N containing 0 and the points zn,

then the series defined by (3a), (4) will converge to f in a certain subregion

of N bounded by a level line of the G r e e n ' s function of B with pole

at z = 0. This result has been established using the convergence properties

of certain sets of rational functions introduced by W al s h [14, w 8.8]. This

result thus shows the power of the method of interpolation series for a

multiply connected region, and the sets derived from (6), (7)are remarkable

in that they simultaneously possess R i e s z - F i s c h e r as well as T a y I o r

structures. Furthermore, they are of particular interest in that they yield

effective means for the representation by interpolation of functions analytic

in multiply connected regions. O)

The biorthogonal functions c?,*(z) obtained from the preassigned set

of functionals {L,} are not elementary, but may be expressed in closed form

in terms of the B e r g m a n [3] kernel function KB (z , w) of the region B,

or alternately, in terms of the G r e e n ' s function G ( z , w ; B) . This is

done in w 3, and the explicit representations obtained will be recognized to

be related to the early work of E. S c h m i d t on the solvability of systems

of equations in Hilbert Space, and to some later work of B e r g m a n who

considered several specific problems of the type worked out generically here.

Also given in w 3 are an analogue of the Schwarz Lemma for functions of

L 2(B) which satisfy L i ( f ) = 0 ( i = 0 , 1 . . . . . n ) , as well as criteria for

the continuability of f from the class S to the class L 2(B).

In w 4 we consider briefly the general problem of interpolation in an

arbitrary region, while in w 5 we show how the orthogonal functions related

to (7) may be employed to solve a problem of the best approximation to

functions regular in a subregion of B by functions of class L 2 (B). The

then entirely analogous to those of w 4 except that in the present paper we use the norm of H i l b e r t space, and therefore obtain expansions of the special form

c ~ d n ( z ) where the c n but not the d n ( z ) depend on the function expanded. 4. Cir. also work by M a l m q u i s t [10] and W a l s h [13] on interpolation in

the unit circle.


criterion o f nearness is automatically determined by the region B, and

explicit solutions are found in terms o f the orthogonal functions. These

results are companion to some recent work in the best approximat ion of

analytic functions by W a l s h and N i l s o n [16], and D a v i s [6].

2. T h e C o n v e r g e n c e of O r t h o n o r m a l l n t e r p o l a t o r y S y s t e m s .

Let there be given a bounded region B which contains the origin. Further-

more, let there be given a sequence o f points {c~k} interior to B for which

we have (s)

(8) lira ak = 0 . k-->-oo

For each integer n (n = 1, 2 . . . . ), we consider a problem P , as fol lows:

o f a l l f u n c t i o n s O ( z ) o f c l a s s L2(B) w h i c h s a t i s f y ( e )

(9) ~ ( a l ) = 0 , O(Ct2)= 0 . . . . . ~ ( % - - 1 ) = 0 , O ( C t n ) = 1 ,

d e t e r m i n e t h a t f u n c t i o n f o r w h i c h

(10) II ~ 1[ = ( O , ~)1/2 = minimum.

Let (p~ (z) designate the unique solution to the problem P , . By writ ing

(11) = IIq .ll ( " = * ' 2 . . . . )

we obtain a system which is complete and or thonormal for B. (Quest ions

as to the existence, uniqueness, completeness, or thogonali ty as well as the

representation o f the r (z) have been deferred to w 3, where these matters

will be taken up in a general way.) W e shall now show that the system

(11) has the property that if f ( z ) is regular at 0 and at all poin ts a n ,

then a series

5. More general conditions as to the distribution of the a k can be imposed. We consider (8) for the sake of simplicity.

6. We use here and below the phraseology that is applicable to d i s t i n c t points a k, Nevertheless, all the reasoning when suitably modified is valid even if the points *t~ are not distinct. In the latter case, if a n occurs precisely v times in the set a~, a 2 . . . . . an_ 1 , equations (9) shall be interpreted as requiring q~ (z) and its first v - - 1 derivatives to vanish in a n, while cp(v)(an)=l. A set of conditions of interpolation f(gk)----3k, k = l , 2 . . . . . n , is interpreted as prescribing f(z) and its first v - 1 derivatives in a point gk of multiplicity-v; this interpretation applies in particular to the formal expansions (15) and (19).


(12) f(z) ~ ) , a. ~." (z)

may be defind by interpolation to f which will represent f in an appropriate

subregion of B.

W e require at first an extension of the Schwarz Lemma which is

obtainable from the principle of maximum modulus.

L e m m a 1. Let f (z) be analytic and in modulus <=M in a region B"

containing the points ~1 . . . . . (~,-1. If, now, f(a~)= 0 for i = 1, 2 . . . . . n - - i ,

then we have

(13) If(z)[ ~ Mexp[-- E G ( z , 5 , ; B')], z~B'. L ~ J

Here G ( z , w; B') designates the Green's function for the region

B ' with pole at z = w . W e omit the p roof of Lemma 1. In addition, we

shall require as a lemma, the following result o f W a I s h (7) which we state

in full for convenience of reference.

L e m m a 2. Let B2 be a bounded region containing 0 whose boundary

b2 consists of a finite number of.mutually disjoint Jordan curves, and let

the points a~ satisfy (8) and lie in B2. Then there exist points ~i on b2

with the property

(~-~0 ... 0-~) ]~/" (14) n-->-o~lim (z - - ~3 0 (z - - [3,) , = exp [ - - G (z , 0 ; B,)]

uniformly in any closed set B'2 C B2, where B'2 does not contain 0 nor

any ai. I f the function f (z) is regular at the points ai and in the region

C~ (B2) : G (z , 0 ; B2) > - - log r >= O, then the expansion

(1~) f ( z ) = ao + as ( z - ,~O ( z - , ~ , ) ( z - ~ . ) (z-l~O +a~ (z- i~O(z- /~ . ) + . . . .

where the coe~cients ak have been found by formal interpolation at the

points z = ai, is valid in cr(e2) and uniformly on any closed subset.

Furthermore, i f f (z) is regular in the region C,(B2), O<r< 1, but is

regular in no region C~, (B2), r < r ' < 1, then we have

1 (16) tim sup[a, , I */n = - - ;

n--).<x~ r

7. Cf. W a l s h [14, pp. 205, 214--217].


i f f (z) is regular throughout B2, the first member of (16) is not greater

than unity.

With regard to (14), a word of explanation is in order. From W a l s h - - t

[14, p. 215] we may find points ~ on b2 such that uniformly in every B2

Z n I Un

(17) lira (z - - ~,) -- (z - - ~,) = exp [-- G (z, 0 ; B2)]. n.->-oo

But from (8) it is clear that l o g [ z - ~, ] ->- log lz ], so that by arithmetic-

mean summation of the sequence we have uniformly in B2

(18) li+m ~ - E l o g ] z - 0 t i l - l o g l z I. /2 k----l

The relation (14) now follows.

It is clear by inspection that the expansion (15) found by interpolation

to f ( z ) in the points ctk is unique, and the determination of (15) requires

that f ( z ) be defined, but not necessarily regular in the points ctk not in

Cr (B2); of course if f ( z ) is defined at a point Ctk it may be considered

analytic there for our present purposes, for f (z) may be defined as iden-

tically constant throughout a suitable neighborhood of ctk. If the definition

of f (z) is altered in points o.k not in C, (B2), but is not altered elsewhere,

the coefficients ak of (15) are correspondingly modified, but the validity in

Cr(B2) of the expansion (15) as modified persists. Similar remarks apply

to the expansion (19).

We proceed to prove our main result for the set {cp~ (z)} defined

by (11).

Theorem 1. Let B be a bounded region containing the origin, whose

boundary b consists of a finite number of mutually disjoint analytic Jordan

curves. Let there be given a sequenc3 of points {c~,} lying in B and for

which (8) holds. I f the function f (z) is defined for every z = r and is

analytic in the rejon C,(B), 0 < r < 1, then the expansion in orthogonal

functions

(19) f (z) = ~ , ci ep* (z), i = l

where the coe~cients ci have been determin,d by formal interpolation in

z =: r is valid in C~ (B), uniformly in any closed subset.


Proof. It will be remarked at the outset that the hypothesis that b

consists of analytic curves is unnecessary, and in a paragraph which follows,

we shall show that the theorem is valid with arbitrary Jordan curves

replacing analytic curves.

Let f ( z ) be regular in the region C,(B), 0 < r < l . Inasmuch as b

has been assumed to consist of analytic curves, we may extend G (z, OrB)

across b to a region Bz, BCB2, whose boundary b2 consists of the level

curves G ( z , O ; B ) = - - I o g r 2 , r2 > 1, and whose Green's function is

G ( z , O ; B z ) - - G ( z , O; B) +logr~. We now construct the set of rational

functions

(20) G~ 0 ) = ( z - - al) ... ( z - - a~) (k = 1, 2 . . . . ) (z -- ~,) ... ( z - - ~ ) '

G o = 1,

described in Lemma 2. The points ~ have been selected so as to lie on bz,

Each function Gk(z) is regular in B2 and is a fortiori of class LZ(B).

It therefore possesses the Fourier expansion

(21) " G~ (z) = c~. 9 . (z) c~. n0.') ,

converging absolutely and' uniformly in every closed subset of B. However,

in virtue of (20) and (9), the first k coefficients in (21) must vanish, so

that we have

G~(O = ~ , c , , qd (z ) , ( k = 0 , 1 . . . . ). ~,=/~ + 1

(22)

Now,

(23) T~ ).s tc,,l ~= f f lG~(z) l~dxdy, •--------k+l B

so that in view of (14), to every ~2>0, there exists a constant m2 such that

(24) rk <= m2 + E2 , (k = O, 1 . . . . ).

By Lemma 2, f ( z ) possesses an expansion

(25) / ( z ) = ~ b, ~ , ( z ) , /~-=0

j. L. WALSH and PHILIP DAVIg

with

(26) lim sup l bkt t/k < r j_~ k--~-oQ r

(for we note that C, ( B ) = C,, (B2), r* = r ~ ) . Thus we have

(27) f (z) = ~, bk s ck.q~*(Z). k=O n : k + 1

The order of summation may be inverted to yield the series

(28) / ( z ) = d91~(z), a91= b~c~., ~ - 1 k=O

which is of form (19), provided that

k=0 n:-k + 1

Now let O<r"<r<r'< 1. Each cp~(z) is regular in the region B ' = C,, (B),

the expansion of the kernel K~(z, ~) = ~J,e:(~)]2 converges C I D

a n d since

uniformly there, we have n=l

(30) t~.*(z) l<M; z~B', ~ = ~ , 2 . . . . .

By Lemma 1, we have therefore in B'

n--1

(31) ]q~91(z)[ ~ Mexp -- G(z, cq; B' .

Now we obviously have log Jz-- cql ->" log lzl uniformly in any closed

region B'0 consisting of B" less a neighborhood of O. The functions

G (z, c~i ; B') + log I z - - ~1 are harmonic in B' and approach the function

G (z, 0 ; B') + log]z] uniformly on the boundary of B', whence

(32) e ( z , ~,, ~ ' ) + l o g l z - - ~ g l + G ( z , o ; B') + log lz l

uniformly in B'. If we consider the arithmetic means of these first members,

we have 91--1

(33). n-.o~lim t[ 1-~-- ~ ": ,~,; B')] = G(z, o', B')= G(z, O', B ) + l o g r '

uniformly in B~ except in the neighborhoods of 0 and the points c~i. Then

by (31), we may write for z on the locus G(z, O; B) = -- logr" (assumed


to pass through no ai), and hence uniformly for z in C,,, (B) r "

lim sup [q3*(z)]l/" < - - n-->-oo

so that to a given 83 > 0, there exists an m3 such that

Iq3n(Z) l ~" m3 + ~3 ; zECr , , (B) , n = 1, 2 . . . . .

Thus we have

(34) [bkl ~, ]ck,,q~:(z)' S- Ibk'( ~ lck,,'2) '/~. (~[+llq3:(z)'2) t'2 n = k + l n n _

With r~<r fixed, it is clear by selecting ~1, .~2, ~3, and r' sufficiently close

to 0 and 1 respectively, that

so that (29) is valid and hence (28) converges uniformly and absolutely

in C,,, (B).

The expansion (28) is unique, for since (28) converges to f(z) in

C,,, (B), the coefficients dn must coincide with the coefficients cn of (19)

obtained by interpolation to f (z) in all the points z = ai. We emphasize

the fact that the connectivity of the subregion of B in which (19) holds depends

upon the largest region Cr(B) in which f(z) (and its analytic extension) is

regular. When B is fixed, this connectivity varies with f(z) and need be

neither simple nor the connectivity of B itself. Again, we should notice

that the expansion (19) in orthonormal functions is valid throughout B when-

ever f is regular in B whether or not f is of class L 2(B). Corollary. L e t f(z) be r e g u l a r t h r o u g h o u t C,, 0 < r ~ l ,

b u t t h r o u g h o u t no r e g i o n C,,, r < r ' < l ; t h e n f(z) p o s s e s s e s

an e x p a n s i o n o f f o r m (19) f o r w h i c h

(35) lim sup l cn l '/~ - 1 r n-->.oo

I f f(z) is r e g u l a r t h r o u g h o u t B, t h e n (19) is v a l i d a n d t h e

f i r s t m e m b e r o f (35) is n o t g r e a t e r t h a n u n i t y .

10 J.L. WALSH and PHILIP DAVIS

C o n v e r s e l y i f a s e r i e s (19) is g i v e n f o r w h i c h (35)

h o l d s , 0 < r < l , i t c o n v e r g e s u n i f o r m l y in e v e r y C,,, r '<r;

b u t n o t u n i f o r m l y in any C w , r " > r ; t h e sum o f t h e s e r i e s

is a n a l y t i c t h r o u g h o u t t h e i n t e r i o r o f C, b u t n o t t h r o u g h -

o u t t h e i n t e r i o r o f any C,,,, l > r " > r ; t h e f u n c t i o n f ( z )

r e p r e s e n t e d is d e f i n e d at e a c h ~i a n d h a s t h e g i v e n s e r i e s

(19) as i t s f o r m a l e x p a n s i o n f o u n d by i n t e r p o l a t i o n in

t h e ctr I f a s e r i e s (19) is g i v e n f o r w h i c h the f i r s t m e m b e r

o f (35 ) i s n o t g r e a t e r t h a n u n i t y , i t c o n v e r g e s u n i f o r m l y

in e v e r y C ~ , r < l , t h e sum o f t h e s e r i e s is a n a l y t i c

t h r o u g h o u t B, a n d t h e f u n c t i o n r e p r e s e n t e d h a s t h e

g i v e n s e r i e s (19) as i t s f o r m a l e x p a n s i o n f o u n d by i n t e r -

p o l a t i o n in t h e ai. T h u s i f f u n c t i o n a l v a l u e s a r e g i v e n

in t h e p o i n t s cti, a f o r m a l d e v e l o p m e n t (I9) e x i s t s ;

e q u a t i o n (35) w i t h 0 < r ~ l is a n e c e s s a r y and s u f f i c i e n t

c o n d i t i o n f o r t h e e x i s t e n c e o f a f u n c t i o n f ( z ) a s s u m i n g

t h o s e v a l u e s in t h e p o i n t s ~i, a n a l y t i c i n t e r i o r to C~ b u t

n o t a n a l y t i c t h r o u g h o u t any C~, , , l~ r">r ; t h a t t h e f i r s t

m e m b e r o f (35) be n o t g r e a t e r t h a n u n i t y is a n e c e s s a r y

a n d s u f f i c i e n t c o n d i t i o n f o r t h e e x i s t e n c e o f a f u n c t i o n

f ( z ) a s s u m i n g t h o s e v a l u e s in t h e p o i n t s czi, a n a l y t i c t h r o u g h o u t B.

We now show how it is possible to remove the hypothesis of the

analytic boundary used in the proof of Theorem 1. To this end, we shall

require the following lemma.

Lemma 3. Let B be a bounded region containing the origin and

bounded by a finite number m of mutually disjoint Jordan curves. Let

C,(B) denote the locus in B, G ( z , 0 ; B ) = - - l o g r , r fixed, O < r < l .

Let them be given an arbitrary r * : o < r * < r < l . Then there exists a

region B* bounded by m mutually disjoint analytic Jordan curves such

that BCB* and the locus G O , O ; B * ) = - - l o g r * in B* lies in B in the region G(z , O; B ) > - - l o g r .

P r o o f : Choose a monotonic sequence of regions B1, B2 ..... converging

to B, each Bk being bounded by m analytic Jordan curves, precisely one in

INTERPOLATION AND ORTHONORMAL SYSTEMS 11

each of the Jordan regions which form the complement of B , the closure

of Bk shall lie in Bk--1, and the closure of B (but no other point) shall

lie in all B, . The function G ( z , 0 ; B) or G ( z , 0 ; Bk) may be defined

as the function - - l o g p plus the function harmonic interior to B or B , ,

continuous in the corresponding closed region, and whose boundary values

are l o g p , p = ( x 2 +y2)t/~. It will be noted that these boundary values are

independent of k, and are continuous in the neighborhood of the boundary

of B. It therefore follows by the classical result of L e b e s g u e [8] on the

continuity of harmonic functions in variable regions, that we have

(36) lim G (z, 0 ; B,) ~ G (z, 0 ; B) k--)-oo

uniformly in the closure of B. Then for k sufficiently large we have on the

boundary of C, (B) the inequality

(') (37) G ( z , O ; B k ) < G(z ,O; B ) + l o g ~ - = - - l o g r * .

Consequently, the locus G(z , 0 ; Bk) --- -- log r* lies in the region

G(z, 0; B ) > - - l o g r . We choose B* as a B~ with suitably large subscript.

Theorem 2. Let B be a bounded region containing the origin whose

boundary b consists of a finite number of mutually disjoint Jordan curves.

Let there be given a sequence of points {as} lying in B and for which (8)

holds. I f f (z) is defined for z = ai and is analytic in the region C, (B),

0<r_<__ 1, then the expansion (19) in orthogonal functions where the

coe~cients have been determined by formal interpolation in z = ai, is valid

in C,(B), uniformly in any closed subset.

Proof. As before, we select O<r"<r ~ 1 and prove absolute conver-

gence in C,.,(B). Select also r* with 0 < r"<r*< r ~ 1, and by the lemma

just established, let B* be a region bounded by analytic curves such that

B CB* and C,,(B*) lies interior to C,(B). Let ~*(z) designate the set of

orthogonal (interpolation) functions for L2(B*). Let f ( z ) be regular in

C, (B). By Theorem 1 we have

(38) f (z) ---- ~ , as ~ (z), f ;~-I

holding uniformly and absolutely in C~, (B*), and moreover,

12 J. L. WALSH and PHILIP DAVIS

1 (39) lira sup[an [Un ~ r__ ~-

R->-oo

Now each (I)* (z) is of class L 2 (B*) and afort iori of class L 2 (B). It there-

fore possesses an orthogonal expansion 03

= ~ " = (~" , ; ) (40) ~; 0) Cnk q ~ ( z ) , c.k , , k=n

the first ( n - 1) coefficients of (40) vanishing because of (9). As before,

by substituting (40) in (38) and rearranging the terms, we shall obtain an

expansion of f(z) in terms of q~k (z). This rearrangement is valid provided 00 ~_~ tt ,

(41) ~ l a . I ~ I c.~ ~ (z) ~< ~ .

, ( r ~ )n As before, we have by (33) for given ~ ( > 0 ) , tcgn(z)l ~ m~ ~ + r 7

z~c,,,(B), o<r"<r'<x. Now ~lcn~l~= ffl'~',t~dxdy---- 0(1) for B

k ~ n

(n-~ oo). Whence

( ')" By (39), [a,l <m~ ~, + ~ - . Hence (41) will hold in C,,,(B) if

( +')(.-) ~ ) - ~ + -~- ~ 1. This can be achieved by taking ~ and e~ sufficiently

small and r' sufficiently close to 1.

When the points a,, are assumed to coincide at z = O, (cf. (6) ) ,

Theorem 2 continues to hold for the corresponding set q~n* (z). In this case,

qon(z) is that function of class L2(B) satisfying

( 42 ) ~ (0) = ~ ' (0 ) = . . . = ~Cn--,~ (0 ) = 0 ; q0~"~ (0) ---- 1 ,

for which I[cpll is minimum. Our theorem now asserts that if f (z) is

regular in any neighborhood N of the origin, then the series (19) obtained

by formal interpolation at the origin converges uniformly and absolutely in

any C,(B) whose closure is contained in N. The functions q~(z) cor-

responding to (42) were originally considered qua orthogonal functions and

without Theorem 2, by B e r g m an. It should be remarked here that when

the region B is simply connected, the functions q~,(z) satisfying (42)are


given explicitly by

( 4 3 ) ~ , (z) - h" (z) [h (0]" n !

where w = h(z) , w ( 0 ) = 0, w ' ( 0 ) = 1 is the mapping function of B onto

a circle. For this case, the possibility of expanding functions analytic in a

subregion of B in a series of cpn is trivial. However, as we have seen, our

result holds also for the multiply connected case as well as for the more

general situations of Theorems 1 and 2.

Under the hypothesis of Theorem 2, the precise extension of the

corollary to Theorem 1 is valid. The proof is left to the reader.

The following asymptotic expression is of interest. Here we have

assumed that the points ai are distinct.

Theo rem 3. We have under the hypothesis of Theorem 1

lim sup q;,*(ct,,) ~1" == ko, n-~oo (~.-~0 ( ~ . - ~ - ~ . (~.- ~,-,)

(44)

where

(45) k0 = lim exp [ - - G ( z , 0 ; B) - - log lz I]. g--~-0

Proof. As in the proof of Theorem 1, let B ' = C~,(B). Here r ' < 1,

and is taken sufficiently close to 1 so that alI points z = a i Iie in B'.

From (31), we have in B'

] �9 S I ~ I

( z - ~ l ) . . . ( z - ~ . _ , ) = _ '

If we set

(46) Xi (z) = C (z, ai; B') -{.- log I z--c q I - - log Izl - G (z, o; B) - - log r ' ,

then by (32) and the last equality of (33), we have Ki(z)--~ 0 uniformly

in B'. Now,

cp:(a,) 1/ .< M1/nexp[ 1 , - s (~,, - - ~,)- 75.. ~ . - - ~ . - 1 ) = {K, (a . ) + C ( ~ , , 0 ; B )

+ logl anl + l o g / ) ] .

Thus

lim sup q~ (co,) 2/, < k0


and since r ' < 1 is arbitrary, we have

(47) lim sup I

/

(.,) 1/n

n-~oo ( ~ -- a 0- .._ (a-~ -- ".-1) ~ k0.

To establish the inequality in the opposite direction, we note that by

the minimal properties of the functions q~, we have in the notation of(20)

(48) I O,_,(a~) I ~ I+P'("-)l

Here the poles of G,,(z) are assumed to lie on the boundary b2 of /32.

In view of (14) and the fact that exp [ - -G(z , 0; B2) ]< l for z in /3,

we have ]G,(z) I<M1, z on b, ( n = 0 , 1 . . . . ) . Therefore IIG~(z) IIB<=M2 ( n = 0 , 1 . . . . ). Thus,

I G._, (.) 1/~<1 ~:(a.) I ~'" (49) M2(a._a~)(~Z_ff-2)_." (a.--a._t) = (a.--al) 2.-(-a~--a~_,)

The left-hand member of (49) reduces to [M2 (am--[31)... (a+--ft ,_l)] -1/' ,

whose limit c8) as n->-oo is ko(/32). By allowing /32 .a~./3, inequality (47) is

established in the opposite direction, and (44) is proved.

Similar asymptotic expressions are valid when several or all of the

points ai are coincident. In the latter case, the following may be proved.

Let H be conjugate to G ( z , 0 ; B) and set ~ ( z ) = e x p [ - - G - - i l t ] ,

k l = [~ '(0)[ . Then,

(50) lim sup[ Iq):(')(~ 1'/" n--~<~ L ~t ! : kl .

3. Basic Proofs and Representat ions.

In the present section we adopt a fairly broad point of view and

discuss generically the solution to problems of which P,, is a typical example.

We shall suppose, then, that there has been given a set of linear functionals

{L,I (n ~ 0, 1, ...) defined over L 2 (/3) and satisfying the following conditions :

(a) E a c h L , is b o u n d e d o v e r L2(/3).

(b) T h e s e t i s i n d e p e n d e n t . T h a t is, fo r e a c h n ( n = 0 , 1 .... )

t h e r e e x i s t f u n c t i o n s fo, f l . . . . , f , EL2(/3) s u c h t h a t t h e

G r a m d e t e r m i n a n t d o e s n o t v a n i s h :

8. Use (8) and W a I s h [14, pp. 214--215].


(51) [L~hlt,,i=0., . . . . . . ) = # o.

F i n a l l y , in s o m e c a s e s , we s h a l l r e q u i r e

(c) T h e s e t i s c o m p l e t e i n L 2 ( B ) o r S. T h a t is, t h e

c o n d i t i o n s

(52) L . ( f ) = 0 ( n = 0 , 1 . . . . ) ,

i m p l y t h a t f v a n i s h e s i d e n t i c a l l y , f in L2(B) o r in S.

The functionals Ln with which we shall be dealing will be thought

of as having definitions which are i n d e p e n d e n t o f t h e s p a c e L 2(B)

and which are a p p l i c a b l e t o a w i d e r c l a s s o f f u n c t i o n s S.

Thus, for example, each L, may be a differential or a "point" operator,

applicable therefore to all functions which are analytic in an appropriate

region; or they may be given by an inner product rule in a wider Hilbert

Space S totally unrelated to g 2 ( a ) . We should remark here that in contra-

distinction to the case of functions of a real variable, the usual point and

differential operators of interpolatory function theory are bounded operators (9).

As examples of sets of functionals for which (a), (b), (c) hold, we may

cite (6) and (7). Others are readily supplied and will appear in the sequel.

According to a classical result of F. R i e s z , each bounded linear

functional L has an inner product representation

(53) L ( f ) = ( f , l)

for some l (z) E L 2 (B) and for all f E L 2 (B). Having been given a bounded

linear functional L (whose scope may extend beyond Z 2 (B)) , we may find

the l ( z ) associated to L in the sense of (53) in the following way. Let

{gn (z)} be any closed orthonormal set for L 2 (B). For f E L 2 (B) and z in B,

2 (54) f 0 ) = a . ~ . ( 0 , a. = ( f , ~,,), a.i~<o o, n ~ 0 n = 0

so that by (53), L ( f ) = angn(z) , l ( z -~ • a.L(~.(z)) . n ~ 0

Since this

is valid for all f ~ L 2 (B) , the last series must converge for all a~ with

~ la,,17<ee. This, by a well-known result of L a n d a u , implies that n~-O

9. Cf. D a v i s [5].

16

~ f L (g . ) I ~ < ~ .

n--"~-O

J, L. WALSH and PHILIP DAVIS

Again, by the K i e s z - F i s h e r Theorem (to), the function

w in B, is of class L2(B) , and may be identified with

[Lz KB (z, w)] - . Here K B ( z , w) is the (Bergman) kernel function for the

region B , Lz indicates that the operation is to be carried out on the vari-

able z, and [x + i y ] - ~ x - - i y . It is now easy to see that

(55) L ( f ) : ( f ( w ) , [LzKB(z ,w)] - ) , I E L Z ( B ) ,

so that we have

(56) l (z) ~ [Lz 1 ~ ( z , ~ ) ] .

As far as functions o f f (z) E L 2 (B) are concerned, we may express L ( f )

through (53) and (56).

W e now consider the problem o f finding a set o f functions

{qbn*(z)}-----{q~(z; L)} which are closed and orthonormal with respect to

L 2 (B) , and are simultaneously biorthonormal to a fixed (but not preassigned)

set o f linear combinations of L . in the sense

L , ~ C ~ ' ) - - & . . , ( m . n = o . 1 . . . . ) (57)

where

for some constants ain. This problem may be solved by setting up a sequence

of minimum problems Qn (dependent upon {Ln}) as follows. F o r e a c h

f i x e d n o n - n e g a t i v e i n t e g e r n, l e t u s d e t e r m i n e t h a t

f u n c t i o n wnEL2(B) s a t i s f y i n g

(59) L0 (,~.) = L, (,~.) = . . . --- L , - 1 ( , ~ , , ) = o ; L,, (,~,,) = 1

a n d w h i c h m i n i m i z e s ttqv,,/[. The following theorem summarizes the

situation.

T h e o r e m 4. Under conditions ( a ) a n d (b) above, the problem Q, has,

for each integer n, a unique solution c9, (z). The set of minimal functions

{q~,, (z)} is orthogonal over B. The set {q~,(z)} is complete in L' (B) i f and

only i f {L,,} is complete in L 2 (B).

10. Cf. B e r g m a n [3, pp. 5--8].


Proof . At the outset, let us note that there surely exist functions of

class L2(B) satisfying (59), for, we need only take an appropriate linear

combination of the functions f , , (z) whose existence is guaranteed by (b).

The existence of a minimal function of class L ~(B) may now be proved in

the usual way by a normal families argument. The limiting function obtained

in this way satisfies the conditions (59) because of (a).

N o w let q),,{z) be any solution to problem O,, and let f e L 2(B)

satisfy the conditions

(60) L0 ( f ) =: L, ( f ) --- . . . ----- L,, ( f ) = O.

Then we assert that (qJ , , f ) = O. For suppose not. Consider the function

cO'= q,, + cj'. For each choice of c, c 9' satisfies (59). Moreover,

I] {p']12 _= ]]ep.+cfll2= I[{p. II2 + c ( / , {p, ,)+ c ({p,, , f ) + Icj 2 I I / l I ~ .

I f we make the selection c = - - ( f ' q~") , then it will be found that [ifH ~

Jl q~' I]~ = II ~ . ]] s [('P"'/)]~ t l f l l ~ < l ] ~ , l / ~ -

This contradicts the assumption that {Pn has the minimum norm among

those functions satisfying (59).

It follows from the argument just presented that the set ~,c9,, I is

orthogonal over B, and moreover, that each %, is unique. For suppose that

cp~ and c9', were both solutions to Q,~. Then f = c9~--cp', satisfies (60) so

that (%, , % , - - q)',) = O. But similarly, (cpn , % , - - q)',) = O, so that I I

L I ~ , . - ~-I t = 0 , w h e n c e ~ ---- ~ . . W e note in passing that because of the condition L,,(tp,J-~-1, we

cannot have qOn= 0 . Hence, setting II q}-tl - k , , > 0 , the functions

(61) , e : ( z ) - - ' P " ( ~ ) 0 , - - o , I, 2, ...) k.

will be or thonormal over B.

As regards completeness in L2(B) let (c) hold. For

construct the function

(62) g(z) ----- avq~v(z), av := ( f , qP*). v z O

Completeness will follow if we can show f ~ - g . To this end,

f e L 2 (B) ,

determine


a function

(63)

such that

(64)

n

f . (z) = ~ a~. w; (z) ~ 0

t i ( f - / . ) = o ( j = o, 1 . . . . . n ) .

This is possible inasmuch as the Gram determinant

I L , (~ ; ) I~-~ = (k0 k, ... k . ) - ' ~ o .

By the observation following (60), the condition (64) implies ( f - - f n , qo*.) = O, n

so that av = avn. Therefore, L i ( f (z) -- E a v q0~ (z)) ---- 0 ( j = O, 1 . . . . . n). V ~ 0

By condition (a), this implies that L s ( f - - g ) = 0 ( j = O, 1, 2 . . . . ) and

hence by (c), f ~ -g .

Conversely, suppose that {q0,* (z)} is complete in L z (B ) . For f ~ L2(B)

we have

E (65) ] (z) = a. q~. (z) ; a. ~- ( f , c?*). n ~ 0

By (a) and (59) there is obtained successively,

to i f ) = ao Lo (~o) ,

L, U ) = ao L, (qOo) + ~1 L, ( r

(66)

L. ( D -- ao L, ( ~ ) + a, r.. (~,) + ... + ~. r.. (q~.),

so that

(67) a. =

Lo(~;) o o . . . L o ( f )

L1 (q%) L, (q~) 0 . . . L1 (f)

L. (q~o) Ln (cp,) ,, (~z) L,, ( f )

�9 ( k o k l . . . k . ) .

( n = O, 1, 2 . . . . )


Thus, L n ( f ) = 0 (n = 0, 1 ... . ) implies a n ~ 0 (n = 0, 1 . . . . ) so that f ~ 0.

The set {Ln} is therefore complete in L 2(B). This completes the proof of

Theorem 4.

We turn now to the investigation of (57). Let us think of the deter-

minant in (67) as expanded according to the minors of its last column,

and denote an = an ( f ) by

(68) L~ (f) = ~ . A,, L, ( f ) , i = 0

where Ain. (kok~ ... k,,) -I are the appropriate cofactors. From (67) and (65)

we have, for f E L 2 (B),

(69) L~ (f) = ( f , q~*).

In particular we have

( 7 0 ) " * , " .

The functions q~* (z) are therefore doubly orthogonal, they are orthogonal

in the sense of (2), and in the sense of (70).

It is now a simple matter to obtain the functions W~(z) explicitly.

From (68) and (55) we have

(71) L* (f ) = ~ , Ain ( f (w), [Lo KB (Z, W-)]--) i = 0

n

= ( f (~), ~, a , . i t , , . K ~ ( z , @ - ) i = 0

Comparison with (69) yields

(72) cp*(w) = ~.A,n[L,:KB(Z, w)]-, ( n = l , 2 . . . . ) i = 0

~; (w) = [Lo: K . (z, ~)]- II L 0 , . K . ( ~ , ~ ) j f

or, briefly,

(73) (p~ (w) = [L*,,K, (z, w)]-.

Alternately, we may construct the functions q~(z) by orthonormalizing

the set [Ln,,Ks(z, w)]- (n = 0, 1, ...) over /3.

The formula (73) expresses the functions q~*~(z) directly in terms of

the kernel function of B. Alternately, they may be expressed in terms of


(74)

where

G(z, w; B) by means of the identityO')

KB(z , w) -= -- 202G(z" w; B)

Oz 2 - ~ - x - ' - Oz 2 -o-X-x +

As particular cases, we note that the minimal functions corresponding to

the sets o f functionals (6) and (7) are, respectively, appropriate linear

combinations of

[O'k 'KB(z 'w) I--= ~ ( k = 0 , 1 . . . . ) dz(k)

and of [K,(ctk, w ) ] - - ( k = 0, 1 . . . . ) . An immediate application o f these

ideas is the following theorem.

T h e o r e m 5. Let a set of linear functionals {L,} be given each of

which is applicable to all functions of a linear class S which is larger

than L 2 (a). Suppose that {L, 1 satisfies conditions (a), (b), (c) above, where

(c) now refers to completeness in S. Then a given f ~ S belongs to L2(B)

i f and only i f

(75)

Proof. By (68), the

IL: l 2 < oo . n-----O

L~ are finite combinations of the Ln and hence

are surely applicable to all f ~ S . Suppose first that f ~ L 2 (B) . Then by

(69) it is clear that (75) must hold. Conversely, given an f ~ S for which

(75) holds. Set an ~ L*~(f) (n ~ 0, I , 2 . . . . ) . By the Riesz-Fischer theorem,

the function g (z) = E an cpn (z) is o f class L 2 (B) and L~ (g) = ctn. Thus, ~ - 0

L~ ( f - - g) ---- 0 ( n = 0 , 1, 2 . . . . ) and by (c) this implies that f - - -=g.

Theorem 5 is a wide generalization o f the radius of regularity formula

o f C a u c h y - H a d a m a r d. As a particular case, let us consider the following

situation. Take S as the class of functions regular in a neighborhood of

z ----- 0 and s ( f ) = f c n ) (0) (n ----- 0, 1 . . . . ) . Construct the Ln* accordingly.

Then (75) gives a necessary and sufficient condition on the Taylor coefficients

11. See, e. g., B e r g m a n [3, p. 60].


in order that a function regular at z-----0 be of class L 2 (B).(12) Theorem 5

is, in effect, a method for making this decision given the values L n ( f ) for

a n y comptete set {L~} .(13)

It should be observed that the B e r g m a n doubly orthogonal functions

are included in our discussion of the operators L,* and functions q~. Expli-

citly, let B be a region bounded by a finite number of mutually disjoint

Jordan curves, and let G be a closed region interior to B. Let the functions S

q~0(z), q~l (z) . . . . . be the B e r g m a n functions which are orthogonal over

both B and G, and are normal over B. Then the operators

L~ ( f ) ---- L* ( f ) ~ ( f , w ~ ) ~

are defined not merely for all functions f ( z ) o f class L 2(B) but for all

functions f (z) o f class L 2 (G) ---- S , and the minimal functions corresponding

to {Ln} are precisely the functions q~. Thus the theory of the functions

q0~ (z) is contained in the present treatment.

As a further application of these methods, we derive an inequality

which is in the Schwarz circle of ideas. In the usual form of the Schwarz

lemma, f ( z ) is assumed to be regular and bounded in the unit circle, and

to satisfy f ( 0 ) ~ 0 . In extended forms of the lemma, f is assumed to

vanish at N + 1 points. In what follows, we shall assume that f E L 2 ( e ) ,

for a fixed region B, and is subject to the N + 1 conditions

(76) Ln ( f ) = 0 (n ~ O, 1 . . . . . N ) .

The Ln in (76) are a set of linear functionals assumed to satisfy only (a)

and (b). Through (67) and (72), construct L~* and q~ ( n = 0 , 1, . . . , N)

corresponding to the preassigned set L0, L1 . . . . . LN. In virtue of (67) and

(69), conditions (76) are equivalent to

(77) L*~(f) = ( f , q~*,) = 0 (n = O, 1 . . . . , U ) .

The orthonormal set q%, ... , q~v may be completed by the adjunction o f

certain functions g~, g~ . . . . , and we may therefore write

12. See B e r g m a n [3, pp. 18-20] where an alternate criterion has been given. 13. In apaper to appear, D a v i s and P o l l a k show how Theorem 5 may be

applied to various problems in the theory of analytic continuation. Application will be made, in particular, to the Fatou-P61ya "change of sign" theorem.


(78) g~(~, ~) = ~ ; ( ~ ) ~;(w) + dO)d(,O. i = 0 i = 0

It is clear that the second sum in (78) is a reproducing kernel for the

subspace of functions satisfying the orthogonality conditions (77). For each

such f , we have, therefore, N

(79) f ( z ) = f f {KB(z,w)-- E~*(z)~p*(w)}f(w)dAw. ~ 0

On applying the Schwarz inequality to (79) we arrive at our next theorem.

Theorem 6. If f (z)r L 2 (B) and is subject to (76), then the following inequality holds throughout B,

N

(80) I f (z) l ~ II f II {Ka (z , z) -- E I q~; (z)12} ~12. i'=o

The bracketed member is of course independent of f . In particular,

functions of L2(B) for which Lo(f)= 0 therefore satisfy the inequality

(see (72)) " 2 12

(80a) ]f(z)l<ltfll{KB(-z z)- fL~ = ' II:-.o,.X,, ( z , ~)II 21 "

For the case of the unit circle K B = ax - l ( 1 - z w ) 2 and (80) is the general

form of a type of inequality obtained by G r 0 n w a I 1 and others. (14)

4. Generalized Interpolation Series.

The interpolation series corresponding to a given set of linear func-

tionals {Ln}, assumed to satisfy Ca) and (b) is given by

(81) f (z) oo ~ , L* (f) q~,* (z). n = 0

We assume that the formal series may be constructed for all f ES, SDL2(B). The series (81) is a connecting link between ordinary inter-

polation theory and L 2 theory. For f E L 2 (B), and if the set {Ln} is complete

in F- 2 (B), this series is, in virtue of (69), identical with the Fourier series

for f , and the development (81) represents f in B. This sort of generality,

which is equivalent to completeness, can only be achieved for interpolation

14. See G r o n w a l l [7], L o k k i [9] for further references.


series of polynomials or of rational functions by stringent requirements (15)

on the points of interpolation and poles of the rational functions. For f

belonging to the wider class S, the series (81) may, under certain circum-

stances, represent f in appropriate subregions of B. We have seen examples

of this in w

A generalized interpolation problem fo~ the region /3 can be stated as

follows. G i v e n a s e t o f f u n c t i o n a l s {L.} s a t i s f y i n g (a) and (b),

a n d a s e t o f c o n s t a n t s {13.}. D o e s t h e r e e x i s t a f u n c t i o n f

o f c l a s s L2(B) s u c h t h a t

(82) L . ( f ) = 8. ( n = 0 , 1, ...) h o l d s ? If (82) p o s s e s s e s a s o l u t i o n , we m a y t h e n i n q u i r e

w h e t h e r i t is u n i q u e . Referring to (68), (75) and (82), we see that

a necessary and sufficient condition for existence is that

(83) A.i ~ < ~ .

In such a case, the series (81) obtained by formal generalized interpolation

will converge to a solution of the problem. Furthermore, a necessary and

sufficient condition that the solution be unique is that the set {L.} be

complete, or what is equivalent, that the equality

(84) KBCz, z) = ~ I ~ ( 0 1 '

be valid in B. (Cf. (80)). Another expression of this condition is that

f(~0-----0 for all i, f(z) in L2(B), shall imply f(z)-~ O. Of particular interest is the classical interpolation problem for analytic

functions and the region B,

(85) L~(f)=--f(m) ; a ~ B ( i = 0 , 1 . . . . );

in this case, the functions q~ (z) may be obtained by orthonormalizing the

set Ke (z, ~) (i ~- 0, I . . . . ) over B, and the constants A,a are then obtained

through (67), (68). Under the special hypothesis studied in w we can also inquire of

the existence of a solution to f ( a i )= 13~ (i-----N, N + 1 . . . . ), N sufficiently

large, which is regular only in a neighborhood of the origin. Our next

theorem considers this possibilky.

15. See, e. g., W a l s h [14, pp. 52--64 and 188--212].


Theorem 7. Let B be a region containing the origin whose boundary

b consists of a finite number of mutually disjoint Jordan curves. Let there

be given a sequence of points {a,} lying in B such that lira ~ . = 0, n--~oo

and a sequence of numbers {[3~}. Then, a necessary and sufficient condition

that there exist a function f (z), analytic in some neighborhood of z = 0

and such that

(86) is that

f ( c q ) = ~i ( i = N , N + I . . . . )

(87) lim sup A,,i[~i < oo.

I f the first member of (87) is equal to l/r, 0 < r < l , then f ( z ) is analytic

throughout C, but throughout no C,, with 1 >r' > r and is represented by

its interpolation series (81) throughout C,; i f the first member of (87) is

not greater than unity, then f (z) is analytic throughout B and is represented

by its interpolation series (81) throughout B.

Proof. This follows from (81), (85), (86), (68) and the considerations

of w

We return to the general interpolation problem. Suppose now that

the sets {L,'} (and therefore {q~'}) are not complete in L 2 (B). A solution

to the general problem (82) will then not be unique. If (83)holds , we

have the problem of finding all functions of class Z 2 (B) satisfying the given

conditions (82), as well as the problem of determining that function of

minimum norm. The sets {L,*} and {cp,*(z)} may evidently be completed

(as in Theorem 4) by the adjunction of sets {L**}, ~q~, j each of which

may contain a finite or an infinite number of elements and are such that

the augmented sets are biorthonormal. Each f ~ L 2(B) therefore has the

decomposition cx~

(88) f (z) -- E L* (f) %* (z) + x~ L*" (f) cp** (z) ~ f , (z) + f2 (Z). n ~ O

If by L we designate the subset of L 2 (B) consisting of those functions

which are orthogonal to L, (n = O, 1 . . . . ), then the most general solution

to the problem (82) consists of f l ( z ) , the function obtained by interpolating

formally, plus some g E L . It is clear from (88) that of all the functions


of L2(B) satisfying the given conditions, f l (z) will possess the least norm.

This same conclusion of course remains valid if the L~ ( f ) are finite in number.

In their present form, the criterion (84) for completeness, as well

as the remarks as to the possibility of decomposition, though pertaining

to the theory of analytic functions, must be regarded principally as theorems

for Hilbert spaces, inasmuch as their analytic content is not immediately

apparent. When we deal with the inrerpolation problem (85) for the unit 2~

circle and use the inner product (f , g)c ~ f f gds, then the whole discus- 0

sion is known. 06} Indeed, inasmuch as the (Szeg6) kernel function for this

region is (1 - - zw) -1 , it is easily shown through (72) and (59) that the

minimal functions q~* (z) must have the form

(z - < ) ( z - ... ( z - ( s 9 ) = c,,

(1 - z a l ) ( i - z a 2 ) ... ( i - a S , )

c. an appropriate constant, and the preceding discussion is therefore related

intimately to the Blaschke product. A similar theory for the unit circle

making use of the inner product of the present paper is given by

L o k k i [9a].

5. /Application to Problems of Best /~pproximation.

In the present section we deal specifically with the minimal functions

q~(z) introduced in w which are associated with the functionals L, ( f ) = f (a,).

Condition (8) will be assumed to hold, and the case of coincident points

(6) is also permitted.

W a l s h and N i l s o n [16], and D a v i s [6] have discussed the

problem of best approximating a function regular in a region G by a second

function regular in B, G C B . In the last mentioned paper, functions f ~ L2(G)

were to be approximated by functions f*~L2(B) for which ]If*liB ~ M

in such a way that I I f - - f * l l ~ = m i n i m u m . It was shown that the doubly

orthogonal functions of B e r g m a n played a central role, and that the whole

theory was related to the theory of a certain integral equation of Fredholm type.

As has been explained in w the functions q~(z) are doubly ortho-

gonal (though not necessarily in the Bergman sense). This fact suggests that

16. W a l s h [13], Takenaka [11].


there is a problem of best approximation by functions of class L2(B) which

may be solved by their use. We shall suppose that there has been given a

function f ( z ) which is regular in some region (not fixed) which contains

zero and the points of interpolation {an}. In the case (6) of coincident

points, we assume merely that f ( z ) is regular in a neighborhood of the

origin. Our aim is to develop a theory of best approximation to functions

f ( z ) by functions f* (z) ~ L2 (B) for which I I f * l l a ~ M . Our criterion of

nearness cannot be of the type mentioned above inasmuch as the functions

f have not been assumed to be' regular in some fixed region. We accor-

dingly adopt the criterion

(90) ~ , I L~ ( f - - f*) i s r TM = minimum. n-~.l

Here r is a fixed and sufficiently small number. This criterion means, in

effect, that the deviation of f* from f is small in the vicinity of z = 0.

If B is the unit circle, and if Ln ( f ) = f(n) (0), then (90) can be replaced by

I f (n) (0) __ f,(n) (0)15 r TM (91) n ! (n + 1) ! = minimum.

n ~ 0

If r is selected sufficiently small so that f ( z ) is regular in ]z[=<r then

(91) is equivalent to

(92) f f ]f - - f*[2dxdy = minimum. Izl<r

It should be observed that because of the structure of L*, the criterion

(90) depends in part on the region B. Our principal theorem in this

direction may be stated as follows.

The o rem 8. Let f(z) be regular in 0 and in the points {anl and o~

possess an expansion f (z) = 2 an q~* (z) valid in C, : G(z , 0 ; B) > - - log r,

r < 1 (Cf. Theorem 1). Then the function of class L2(B) of B-norm not

exceeding M which best approximates f (z) in the sense of (90) is given by

a . (93) f* (z ) = k + r 2" ,

where k is the unique positive solution of


] 2 r4n (94) Ira~ + )02

- - m 2 "

Proof . Let there be given a function g (z )E L 2 (B) with

W e have g(z ) = ~ , bn q~, ( z ) , bn = ( g , q~l) = L* (g) and

(95) I bn I s M 2 . ~;--7--.1

Thus criterion (90) becomes

]lg[[B U.

(96) ) ] ] an--b~l s r 2n = minimum.

W i t h {an/ fixed, it is known07) that the solution of (96) subject to (95) a n r 2n

is given by bn = ~ .+ r2 n , where ~ is the posi t ive solution o f (94).

I t should be observed that f * ( z ) is indeed o f class L 2 ( B ) . This is an

immediate consequence o f (35).

R E F E R E N C E S

1. S. B e r g m a n, Zwei Saetze iiber Funktionen yon zwei komplexen Veraenderlichen,

Math. Ann. vol. 100 (1928), pp. 399--410. 2. S. B e r g m a n, Partial differential equations, advanced topics, Brown Univ.

Notes, Providence, R.I., 1941. 3. S. B e r g m a n, The kernel function and conformal mapping, Math. Surveys

vol. 5, New York, 1950. 4. L. B i e b e r b a c h. Lehrbueh der Funktionentheorie, vol. II, Berlin, 1927. 5. P. D a v i s, On the applicability of linear differential operators of infinite order to

functions of class L* (B), Amer. J. Math. vol. 74 (1952), pp. 475--491. 6. P. D a v i s, A n application o f doubly orthogonal functions to a problem of ap-

proximation in two regions, Trans. Amer. Math. Soc. vol. 72 (1952), pp. 104--137. 7. T. H. G r o n w a l I, Some remarks on con formal representations, Ann. of Math.

vol. 16 (1914), p. 72. 8. H. L e b e s g u e, Sur le problSme de Dirichlet, Palermo Rendiconti, vol. 24

(1907) pp. 371--402. 9. O. L o k k i, Ueber analytische Funktionen, deren Dirichlet Integral endlich ist und

die in gegebenen Punkten vorgeschriebene IVerte annehmen, Ann. Acad. Sci. Fennicae, series A, vol. 39 (1947) pp. 7--57.

9a. O. L o k k i, Ueber eine Klasse yon analytischen Funktionen, Comptes Rendus du Dixi~me Congr~s (1946) des mathOnaticiens scandinaves, Kopenhagen, 1947.

10. F. M a I m q u i s t, Sur la d(termination d'une classe de fonctions analytiques par leurs valeurs clans un ensemble de points. Comptes Rendus du Sixi~me Congr~s (1925) des mathdmaticiens scandinaves, Kopenhagen, 1926.

17. Cf. W a l s h and N i l s o n [16], D a v i s [6].


11 S. T a k e n a k a, On the orthogonal functions and the new formula of interpolation, Jap. J. Math. vol. 2 (1925) pp. 129--145.

12. J. L. W a 1 s h, On the expansion of analytic functions in series of polynomials, Trans. Amer. Math. Soc., vol. 26 (1924) pp. 155--170; vol. 30 (1928) pp. 307--332; vol. 31 (1929) pp. 53m57.

13. J. L. W a i s h, Interpolation and functions analytic interior to the unit circle, Trans. Amer. Math. Soc., vol. 34 (1932) pp. 523~556.

14. J. L. W a I s h, Interpolation and approximatmn, Amer. Math. Soc., Colloquium Publications, vol. XX, New York, 1935.

15. J. L. W a I s h, On interpolation by functions analytic and bounded in a given region, Trans. Amer. Math. Soc., vol. 46 (1939) pp. 46--65.

16. J. L. W a l s h and E. N. N i l s o n , On functions analytic in a region; approximation in the sense of least pth powers, Trans. Amer. Math. Soc., vol. 65 (1949) pp. 239--258.

17. D. V. W i d d e r, On the expansions of analytic functions of the complex variable in generalized Taylor series, Trans. Amer. Math. Soc., vol 31 (1929) pp. 43--57.

Harvard University

(Received December 7, 1951)

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Interpolation and orthonormal systems