An Interpolation Problem for Harmonic Functions

An Interpolation Problem for Harmonic FunctionsAuthor(s): J. L. WalshSource: American Journal of Mathematics, Vol. 76, No. 1 (Jan., 1954), pp. 259-272Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2372415 .

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AN INTERPOLATION PROBLEM FOR HARMONIC FUNCrIONS.*

By J. L. WALSH.

The study of functions f (z) analytic interior to the unit circle and defined there by requirements of interpolation is due especially to Kakeya, Takenaka, Malmquist, and Walsh [1] if the functions are subjected to the requirement

(1) f f(z)I2 I dz <oo, C: I z=1,

and is due especially to Lokki [2] if (1) is replaced by the Dirichlet integral of f (z) over I z I < 1. It is the object of the present note to indicate that the extremal methods developed by these authors, and the theory of reproducing kernels developed by Bergman [3] and others [4], apply also to the study of real functions u(z) harmonic interior to C and defined there by requirements of interpolation, where uM(z) satisfies a condition analogous to (1). We indicate the differences of method required for harmonic rather than analytic functions, and present the specific formulas involved.

We denote by H the class of functions (z =ret), 00 00

(2) u(z) = lao + E rk(ak cos cO + bk sin IO), (ak + bk 2) < ?, k=l k=1

each of which is clearly harmonic in I z I < 1. For a given function u (z), the coefficients an and bn are uniquely determined, because on each diameter of C: I z I 1 the function is represented by a power series in the real variable r; such a representation is unique. The series in (2) converges in the mean on C to some function u(e00), and since the partial sums are represented in j z j < 1 by their Poisson's integral taken over C, it follows that u(z) is also represented in I z I < 1 by the Poisson integral of u(e00) taken over C. The norm 11 u 11 of u(z) is defined as the (non-negative) square root of

(3) [4 [u(et0)]2d6 - Iao +E (ak + bk 2) k=1

A function of zero norm vanishes identically. The first series in (2) con-

259 *Received May 12, 1953.



260 J. L. WALSH.

verges uniformly on every circle I z = r < 1, since the ak and bk are bouinded, and we have

t. ~~~~~~~~~~~~00 (4) 1 [u(re00) ]2d = la02 02+ (ak2 + bk2)r2k;

|z|= 2

=

this last member is not greater than the second member of (3), and by Abel's theorem approaches the second member of (3) as r - 1. Of course (4) is valid for any function u (z) harmonic in I z I < 1, so the last condition in (2) can also be interpreted in the form

>2r

(5 ) lim | u (re00) 1 2df <00; r-> I

the integral whose limit appears here increases monotonically with r, as is obvious from (4).

1. An extremal function.

THEOREM 1. With I a I < 1, let fa(z) be the function of class H which takces on the value unity for z = a whose norm is least. Then for an arbitrary function u(z) of class I we have

(6) u(a) u> u (z)fa(z) I dz I/ f.a(Z) I dz 1.

From (2) we have (I an cos nO + bn sin nO 2? an2+ bn2)

00 00

(7) u (reil) < (1+ > r2 ) (iao2 + E (an + bn2) )n 1 1

U 11[2( +2)/ r2)]

from which it follows that a sequence of functions un(z) of class H of bounded norm is normal in I z I < 1. A suitable subsequence converges uniformly on any closed set interior to C to some harmonic function u(z). The monotonic character of the integral in (4) implies

27r ^27r

Tu (reio) l 2dO lim T tn (re 0) 12 dO 7 11 U

11 2; o ~~~~~~~n-> oo

since Un 11 is bounded, (5) is satisfied and u(z) is of class H, with 11 u 11 lim sup 11 uj li. The set of functions of class H taking the value unity in z =- a contains the function u (z)- 1, hence is not empty; there exists a function fa (z) of this set of minimum norm. We have f > 0 by virtue of fa( a) = 1.

LEMMA 1. A function uo(z) of class IS is orthogonal to fa(z) when and only when uo((a) ==O.



AN INTERPOLATION PROBLEM FOR HARMIONIC FUNCTIONS. 261

If we have u0 (ae) 0, we have for every real E by the definition of fa, (z)

(8) 1 fiX 112 I || faC + EUO 2 || fr C|2 ]+ 2E(fC, uo) + E2 || Uo 112,

where we use the generic notation

(f, g) ff ) f z g (z) dz

Since (8) holds for every E we must have (fe, uo) 0. Conversely, if we have (f,' uo) 0 but uo(as) 7& 0, the function

(fa(z) + EUO(Z) )/(1 + EuO (a))

is of class H, takes the value unity when z = a, and has as norm the square root of

(11 fCt || + El || Uo ||21)/ + EU ( a)]2

a norm which for a suitably chosen numerically small E is less than that of fc (z). This contradiction of the definition of fc (z) establishes Lemma 1.

If u(z) is an arbitrary function of class H, the function u(z) - u(as) is of class H and vanishes for z a ., so by Lemma 1 we have

j"[u(z) -u(a)]fu(z) dz = 0;

in particular from u(z) afc, (z) we have 7r 1w fC, 112 = fa(z) I dz # 0, (6) follows, and Theorem 1 is established.

The function fc (z) of Theorem 1 is unique, for if a second such function Fc(z) exists, the function fc(z) - Fa(z) vanishes for z = a, hence by Lemma 1 is orthogonal to both fc(z) and Fa(Z), hence orthogonal to fa(z) -F(z) anad vanishes identically. The function fa (z) of class H is uniquely charac- terized by f(c(a) = 1 and the property of being orthogonal to every uo(z) of class H which vanishes in z = a; for if a second function F4, (z) has these properties the function fc(z) - Fc(z) is orthogonal to both fa (z) and FW(z), hence is orthogonal to fc (z) - Fc (z) and vanishes identically. If ?d (z) is a function of class H and if for every uo(z) in H we have uoQa) 7r(uo(z), D(z)), then the function (z) -fa(z)/7r(fa(z), 1) is orthogonal to every uo(z) in H, is orthogonal to itself, and vanishes identically; thus we have q(z) = fa(z)/-r(fa(z), 1).

Lagrange's multipliers give a necessary condition for a minimum and thereby enable us to compute fc (z). In the notation (2) for fc (z) with o: = peik we have to minimize

on

(9) 2aO2+E (an2 + b 2) 2



262 J. L. WALSH.

subject to the auxiliary condition

(10) lao + pn (a cos no + bn sin no) = .

We thus write (AXu :& 0)

00 00

F(X, p,) - [ao + p pn (a,,, cos no + bn sin np)] + j4[ a02 + E (a.2 + bn2)] 1 1

OF1/0a= Xpn cos no + 21Aan 0, n > 0;

OF/Oao 1 [X + 2tLao] = 0,

JF1/0b =0pn sin no + 2,,ubn ?, n > 0.

Substitution of these unique values of an and bn in (10) now yields 00

- (X/2p) [~ 2+ z p2n (Cos2 no + sin2no)]-1,

-(A/2,(t) 2 (1 -p2) + p2)

lao== (1 -p2)/(l + p2) an - 2(-p2)pncosno/( + p2),

bn-2 (1 p2) p) sin no/ (I + p2)

(11) fci(ret8) = (] -_P2) [(1 - r2p2)/(l - 2rp cos(Q - 0) + r2p2)]/(1 + p2).

The denominator in (6) reduces to 27r(1 -_ 2)/(l + p2), so the second member of (6) reduces to the familiar form of Poisson's integral.

Our entire proof of Theorem 1 can be based directly on Poisson's integral rather than on the general theory of orthogonal functions; the contrast in method -is that of derivation rather than immediate verification, methods of general rather than restricted applicability. The computation of fa.(z) must of necessity yield the kernel of Poisson's integral, for we have already shown that any function (D (z) of class H with the reproducing property

uO((a) = u0(z) L(z) I dz I for every u0(z) in H must be identical with

flu (z)/ 4 fa (z) I dz |. Incidentally, we have here a new proof of the validity

of Poisson's formula. Another form of (11) is (z = re"0)

(12) af (re0) ( p2) Re[ (1 + az)/ (1 - 5Z)](I + p2)

which will prove especially useful.

2. Interpolation and orthogonal functions. We are now in a position to study the general problem of interpolation.

THEOREM 2. Let the distinct points 0, 17 27 . * interior to C be given,



AN INTERPOLATION PROBLEM FOR HARMONIC FUNCTIONS. 263

finite or infinite in number. Let VIfn(z) be the function of class H of minimum norm which satisfies the auxiliary conditions

(13) 1'n (cao) --An (al) * - - n (cn--1) 0, fn (an() =- 1.

Then the sequence {fn (z) } is orthogonal on C.

Theorem 2 is similar to a theorem due to Bergman [3], involving analytic functions and multiple interpolation in a single point of a region.

The set of functions of class H satisfying the auxiliary conditions (13) is not empty, for there exists a polynomial in z satisfying those conditions; the real part of this polynomial is of class H and also satisfies those conditions. Thus +t' (z) exists. No function of class H satisfying (13) is identically zero.

The function 4n(z) defined in Theorem 2 is orthogonal to a function u(z) of class H with uQ(ox) = u(c(1) u(an-,.) = 0 when and only when we have u (,,) = 0; we omit the proof, which follows precisely the method of proof of Lemma 1. Consequently each iln is orthogonal to tn+l, tn+2, , * * . so the sequence is orthogonal.

Moreover, if we have for a function u(z) of class H the equations u (ao) = u (a1) u (an-1) = 0, then we have as in the proof of (6),

(14) U (an) jc u(z) fn (Z) I dz I/, C n6 (Z) I dz 1.

A special case of this equation is found by setting u(z) frn(z)

[+n (Z)]'I| dz qln n(Z) I dz 1

so (14) can be written

(15) u(an) 4 u(z)/n(z) I dz 1/7r 11 gn 112.

If u(z) is an arbitrary function of class H, there are two formal expansions of u(z) of form

(16) u(z) '-'aofo(z) + allfr(z) +

the one expansion is the usual orthogonal function (Fourier type) expansion found by formal expansion of u (z) on C, and the other is an interpolation series expansion found by setting successively z = ?:l, *xl , and solving for ao, a1, * * * in order:

u (xo) aoo (ao), u (a1) - aoio (a1) + ajj (a1)a *

These two formal expansions are identical. The fact that ao is the same



264 J. L. WAL8I[.

whether found by one method or the other is a consequeiice of (15) for it =-- 0. Assume this conclusion true for ao, a,, * * *, an,. The function

Un-1 (z) -u (z) aoko (z) -aljtl (z) * an-jn-i (Z)

vanishes for z ao, a,, , 2n-1. By (1a) it is then immaterial whether an is determined by interpolation to u,-l(z) in z = n, namely an&n(20 ) Un-=(20n

with tfrn(2n) 1, or determined by the formal orthogonal expansion of Un_1(z)

on C: 7ran II fn 11= 4 X un.1(z) tn(Z) I dz 1.

Each function d'n(Z) is unique, and is also the unique function which satisfies the auxiliary conditions (13) and is orthogonal to every u(z) of class H which satisfies the conditions u (ac) = = u (,n) = 0, as follows by the method already used in connection with Theorem 1 for n 0. We shall prove that gin (Z) is a linear combination of the functions 'I (z), J (Z )

' * *, * In (Z), where Ik (Z) fak (z). These functions Ik (z) are linearly

independent on C, for by (12) they are functions harmonic over the extended plane except for singularities in the distinct points /1Xk respectively. These functions *k (Z) can then be orthogonalized and normalized on C by the Gram-Schmidt process, yielding a set 0o (z), f1 (z), , + (z), where Ok (Z)

is a linear combination of the functions '0 (z), (Z), , *kI(Z). Thus 4k (Z) is orthogonal on C to the functions cko, cki , cpk, hence is orthogonal to 01, *1', , ~I7k-1, hence by Theorem 1 vanishes in the points a, i, , a-t. But kk is not orthogonal to 4k, SO (also by Theorem 1) we have I k(Qk) & 0. Every function u(z) which vanishes in all the points (co, cl , k iS orthogonal to *0'1, '1', , 4bk, hence is orthogonal to kk(Z). By our characterization of dt'k(Z) in terms of orthogonality to such functions u(z), it follows that ck(Z() is a constant multiple of k(Z), SO frJk(Z) is a linear combination of the functions 10, *1, , hk; conversely, *k iS a linear combination of io', qj , qk.

The formulas for the orthogonalization and normalization of the functions (12) may be readily written by the reader in determinantal form; we have, for instance

j j(Z)k (Z) I dz |= 4X hj(z)fCk(z) I dz I = jY(Ck) fUak(Z) , dz I

2w[(l - aj 12)/(1 + I aj 12)] [(I _ k 12)/(l + |1 2)]

X Re[(1 + 5jack)/(l jak)].

The two fundamental questions of interpolation for functions of class H are 1) given a function u (z) of class H and a set of points ao~, acj, * y iiiterior to C; to express u(z) in terms of the values U(QXk) ; 2) given a set




of points ,, cl, interior to C and corresponding functional values A,,

Al. ; to determine all functions u (z) of class H satisfying U (ck) = A . The second of these questions is the more general and essentially includes the first, so we proceed to consider it.

There obviously exists a unique linear combination of the functions

~ot(z)-, +1(z), * , tn, (z) which takes on prescribed values Ao, A1, , A, in the respective points cop, sc , ,, so there exists a unique linear combination of the functions 4o(z), I(z), * , nI'(z) which takes on those values in the respective points. The formula for these (equivalent) linear com- binations is readily written in determinantal form. The function uo (z) thus obtained is orthogonal to every function u (z) of class H which vanishes in the points ao, alp, , a,,,, hence is the unique function of minimum norm

which satisfies the prescribed conditions of interpolation:

11 Uo + U II2 11 2o III + 11 U II 2 11 Uo 12.

The case of an infinite number of points ao, a, and prescribed values

Ao, A1, is similarly treated. The points and values define a formnal infinite expansion (16) found by interpolation in the points ak; a necessary and sufficient condition that there exist a function u(z) of class H taking on the values Ak in the points ak iS the convergence of the series E ak2 11 k 1!2.

If this condition is fulfilled, the second member of (16) converges in the mean on C, hence by (7) converges uniformly on any I z ? - r < 1, and represents a functioln uo (z) in H which satisfies the conditions of interpolation. Any function in H which vanishes in all the points ck iS orthogonal to every bk, and conversely; such a function is orthogonal to u0 (z). Thus uo(z) is the unique function of class H of least norm which satisfies the conditions of interpolation. The most general function of class H satisfying the conditioins of interpolation is then u0 (z) plus an arbitrary function u,(z) of class H which vanishes in all the points 2k, or otherwise expressed is u0 (z) plus an arbitrary function u, (z) of class H which is orthogonal to all the Pk(Z).

If we take as departure not arbitrary assigned values Ak but a function u(z) of H, its formal series of interpolation (16) is on C merely the formal expansion of u(e08) (measurable and square-integrable on C) in terms of the orthogonal functions '&k (e0), hence converges in the mean on C and represents in I z I < 1 the function uo (z) of least norm satisfying the conditions Uo (Ak) =U (2) for every ke.

3. Real points of interpolation. There is one non-trivial case where the functions cPk (z) and Vk (Z) can be written down at once:



266 J. L. WALSH.

THEOREM 3. Let there be given the real distinct points

(17) ,, O 21 2, * * ,

interior to C. In series (16) we may choose fro(z) = 1

O,,(z) = Re [,nz (z - a,) *

(z - a,) /{ ( I - az) * lanz) ]

(1) An (I _ 22n) ..(I "2) /2n (n -1) **(2n - 1n-)

The functions

2-2, Re[(- l a j2)-z(z - ) 2 (* * (Z )/{(1- _ 1Z) * (1 _ ( aZ)}]

are norrmal and orthogonal on C.

The analytic functions Frn(Z)- U (z) + iTVn7(z) whose real parts occur in (18) are known ([1], p. 305) to be orthogonal on C; for j=/k we have

0 j F (Z))k(Z) I dz| (UJi + Vj) (Uk- iVk)ds,

(19) 1 (UjUk + VjVk)ds = (UjVk -UkVj)ds 0.

On C we have dz = izds, so by Cauchy's integral formula (even if j k 7 0)

X f (Z))Fk(z)ds -i 4 Fj (Z)Fk(z)dz/z = 0,

(20) f (UjUk- VjVk) ds=f (UjVk + UkVj) ds 0.

The orthogonality of the set exhibited in (18) follows from (19) and (20). The function t n(z) exhibited in (18) is a linear combination wivth real

coefficients of the functions

(21) ~Re[(I + (XkZ) I 2k0)] 7? 0, op 2, * n,

by the classical theory of the decomposition of rational functions; since the procedure of orthogonalization of a given sequence leads to essentially unique results, it follows that the sequence exhibited in (18), found by orthogonalization of the sequence (21), is identical with the sequence f (z) previously considered in (16), found by orthogollalization of the functions Ik(Z) fa,(z) of (12).

From the analogue of (19) with j 7= > 0 and from (20) we have

I Fj 12 ds = 2 Uj2 ds; the first member is readily computed, so nor-

malization is as indicated in Theorem 3.




4. Completeness of orthogonal functions. The functions 7lk (Z) in (16) may be complete but need not be complete. For instance, if the ak can be written in the form k - peikfl, 0 < p < 1, where p is independent of le and where 8/7r is irrational, the points ak are everywhere dense on the circle

I z j p; a function of class H which vanishes in all the points ak vanishes identically interior to C. On the other hand, if the ak satisfy the conditions of Theorem 3, the function Vk (z), kc # 0, in H does not vanish identically and is orthogonal to all the functions t4l (z) [or Un (z)], so the latter do not form a complete set in H.

There are various procedures for completing a given set dt'k(Z) in (16), to which we now turn. An arbitrary function of class H which vanishes in all the points ak iS orthogonal to each tk (z), and conversely. If the functional values Ak iU (cqk) are prescribed with E ak 2 11 'jk 112 convergent, or if a function u(z) of class H is given, the formal expansion (16) converges in I z I < 1 and converges in the mean on C to the function uit (z) of minimum norm with the property uO (ck) =1 U (ck). Every function u (z) of class H which takes on the prescribed values is given by the equation

(22) u(z) guo(z) + u (z),

where u1 (z) is a function of class H which vanishes in all the points ak.

By the orthogonality of u0 and u1 we have

(23) 11 u Il2 u11 7o 112 + 11 U 21 2.

If the set djk(Z) in (16) is not complete, we denote by II' the closure of the linear family of functions u(z) of H defined by the q'k(Z), and denote by H" the closure of the linear family defined by a complete subset of orthogonal functions of H each of which is orthogonal to all the qif; then H" is the class of all functions of H vanishing in all the points 2k, or alternately orthogonal to all the f1. Each function of H is the unique sum of a function of H' and a function of II". It may occur (in contrast to the facts for interpolation to analytic functions, for which the set of points on which all functions of the analogue of H" vanish is precisely the set {k}), and the Blaschke product whose zeros are the ak iS convergent) that each function of H" vanishes on a point set larger than {ak}), as for instance under the conditions of Theorem 3 with ak -> 0; in the notation of the proof of Theorem 3, the functions Fn (z) - Un (z) + iVn (Z) form a complete set with respect to the class H2 of analytic functions E CkZk'% E | Ck 12 < oo; the real part of a function of class Ho is of class H, and an analytic function whose real part is of class H is itself of class H2; thus the functions {U,, V)} form



268 J. L. WALSH.

a complete set with respect to class H; the closure of the linear family defined by {Un} is H', and the closure of the linear family defined by {Vn} is H"; all functions of the latter family vanish at every point of the axis of reals interior to C.

We can be more explicit, with the choice that ck is real, k -> 0. Every function of class H which vanishes for real z is orthogonal to each tk (Z),

hence is orthogonal to each function of class H'. But it follows from (2) that every function of class H is an analytic function of the real variable r on the segment - 1 < x < + 1, y = 0, so every function of class H which is orthogonal to all the qk (Z) vanishes in all the points cXk and vanishes on the axis of reals, hence (by Schwarz's principle of symmetry) satisfies the equation u (z) - u (i). In the expansion (2), which is unique, it follows that every ak vanishes. Consequently, the class H" is precisely the class of functions (2) with every ak equal to zero, namely the closed linear extension of the class {rk sin kl0}; otherwise expressed, the class H" is the subclass of H for which u(et0) is an odd function of 0.

We return to the situation of arbitrary ak where the set H" is not empty, and denote by S the set of points interior to C on which not all the functions of H" vanish. Precisely the interpolation method used in the proof of Theorem 1 yields a formula analogous to (6) for the determination of the value in an arbitrary point a of S of an arbitrary function of class H". Moreover there can be determined by the methods already used a set of orthogonal functions I'to, +",, all of class H" such that an arbitrary function u" (z) of class

H" can be represented

(24) u" (z) .- ao0.'0, + al,"' (z) + **-

the function 1"k (Z) vanishes in points " , a"-l of S and does not vanish in a"k; the unique formal expansion (24) may be found either by the usual method of expansion on C in orthogonal functions or alternately by formal interpolation to u' (z) in the points a"k; the points a"k are not uniquely determined, but may be chosen for instance as the subset in S of the points peik$, 0 < p < 1, k = 0, 1, 2, * , where 8/37 is irrational, and p is independent of k. The functions /"k are all orthogonal to the original set fk, so an arbitrary function u (z) of class H has the expansion in orthogonal functions

(25) u(z) -, aoqo + ajij + + al"o&ob + a",f"'1 +

which is valid in I z j < 1 and valid in the mean on C. Series (25) also partakes of the usual properties of interpolation series; the function +l



AN INTERPOLAITION PROBLEM O1,R01t IJARIMNIWI C FUNCTI1ONS. 2 69

vanishes in the points o, ali, *, .akl but nlot in the point ak, SO if ao, aj, . , 1k l have been determined, the coefficient ak can be determined by fornmal interpolation in the point ak; the function q/'k vanishes in the points 0Yao, *, , o, ai, , k--i but not in the point a"k, so if ao, a,, a a",,

a'', a k--l have been determined, the coefficient a"k can be determined by fornmal interpolation in the point a"k*

5. Kernel functions. The relation of the classes II, II', H" is of particularly simple fornm when expressed in terms of kernel functions. If the set of functions pn(z) each of. class H is nornmal and orthogonal but not necessarily conmplete on C', and if an arbitrary sequence an. with 2 a,2 conver-

00

gent is chosen, the series E anqn(z) converges in the mean on C, and hence 0

by (7) converges uniformly on every closed set interior to -C. Since the series Eann (z) converges for every such choice of the a,, it follows by a result due to Landau that the series , [(.,, (Z) ]2 converges for every z interior to C. Consequently the kernel function

00

(26) lc(z, t) =1 i n(z) ( t) n=0

is defined for each t with I t I < 1 as a function of class II in z, and is symmetric in z and t; the series defining 7c (z, t) converges in the mean for z on C. Tf u(z) is a function of H and if tlle set n (z) is complete in II, we may write for every z with z < 1

, (zy t) u(t) I dt u i nZ U(t) >n (t) I dt u u(z). | C ~~~ ~ ~~n=OC

From this equation it follows that the kernel when defined by (26) in terms of a complete set {p0} of normal and orthogonal functions can also be defined by the analogue of (26) in terms of a second complete set {+'n} of normal orthogonal functions:

1 ' 00

kC(z, t) 4)n (t) dt - -" o(z)n k (z, t! 7- -

AT E t(z+n( *1=0

for I t I < 1, since kl(z, t) is of class H in z for fixed t, I t K < 1; this series for k(z, t) converges in the mean for z on C and uniformly for I z I - r < 1.

A kernel 7k, kc', kc" is associated with each of the classes H, H', H" previously introduced; the kernel can be expressed in terms of a complete normal orthogoni(al set for the corresponding class:



270 J. L. WALSH.

7cf f( Zn to ) . 'tn ( Z ) *"Jntn( )

Since the set (i/n` qt"n} is complete in H, we have k(z, t) = k'(z, t) + k"(z, t). If u(z) is of class II, we have

. '(z, t)u(t) I dt |-=1 E'j(z) U (t) f'n (t) I dt I,

which is the formal expansion of u(z) in ternms of the set {tfr'n}.

If a function u(z) of class H is given, or if functional values in the points ak are otherwise given corresponding to which a function of class H exists, we have a formal development E akf'k(Z) ; if u(z) is given this

development represents the function f k'(z, t)u(t) I dt . If the functions

t'k(z) are complete in H, this development represents the unique function of class H taking on the given values in the given points. If the functions {f 'k(z)} are not complete in II, this development represents the function of class IH of least norm taking on the given values in the given points; the whole class of functions of H taking on the given values in the points ak is precisely the class

00 00 7 aktrk(Z) +Z bk+tf"k(z), where E bk2 < , k=O k=O

and can also be expressed as E akq/k(z) + 7c" (z, t) U (t) I dt |, where U (z) k=O.C

is an arbitrary function of class II. In the special case already considered (? 4) that the ak are real with

ak --> 0, we have shown that IH" is precisely the set of functions u(z) in H for which u(e08) is an odd function of 0; since every function of 0 is the unique sunm of an odd function of 0 and an even function of 0, and since every odd function is orthogonal to every even function, it follows that the set H' of functions of H orthogonal to all functions of H" is precisely the set of functions u(z) of H for which u(e08) is an even function of 0. For this special case we may write (r < 1, p < 1)

k (re0, peOO) = + 4 -1- rnp' cos nO cos no n=1

= 2- Re[1/(1- rpei(O+k)) + rpei(0-)/(11 rpei(O-k))],

1"(re , pek) - - r71p, sin no cos no n=1

2lie [rpei(08)/(1 - rpei(0-0)) - rpei(0+01/(l -pei(0+0)) 1



AN INTERPOLATION T'RO13LEM FOR hIIARMONIC FUNCTIONS. 271

6. Multiple points. In interpolation problenms for analytic functions, coincidence of two points of interpolation corresponds to prescription of the derivative of a function; for harmonic functions, such coincidence corresponds to prescription of a partial derivative of a function; similar remarks apply to coincidence of more than two points of interpolation. For harnmonic functions it is obvious by Laplace's equation that not all partial derivatives of a function at a point can be assigned arbitrarily; indeed, at any point precisely two partial derivatives of any given order can be assigned independently. For- instance, a harnmonic function u(z) which vanishes at z = 0 together with its (n -1) -st derivatives but with alu(O)/lxno0 is

u (z) -lRe (zn) =sXn-n (n- I)Xn-2y2/2 + ;

a harmonic function v(z) which vanishes at z = 0 together with its (n - 1)-st derivatives but with anv (0) /xXn-lay & 0 is vl(z ) - im(Zn) = nx'y- .

The expansion (2) can be interpreted as an interpolation series defined from the values of u (z) and the derivatives Onu/axn and un/Oxn'-lay at z = 0; the corresponding complete set of orthogonal functions of class II is {rn cos nO, rn sin nO}.

The theory of interpolation for harmonic functions as we have already developed it, holds with only minor and obvious changes if partial derivatives as well as functional values are assigned. For instance if z0( j I < 1) and v are given, there exists a function uo(z) in HI with aOu0(z0)/ x" = 1 whose norm is least; any function u(z) in II satisfies the equation Ovu (z, xv == 0 when and only when u(z) and u0(z) are orthogonal; we have for an arbitrary function u(z) of iH

fuo(z) [mt(z) -u0(z)dvu(zo)/x0xv]s = 0,

au((z )/xv -fr uo(z) u(z)ds/lf [uo(Z)12 ds,

a formuila which colresponds to differentiation of P'oisson's integral under the integral signi. The choice u(z) (Tf(z), a multiple of Re[(z-z_O) n

with aOvUo(zO)/1.rv = 1, yieldsj' [uo(z)i)-2 ds { uo(z)U0o(z)ds. We shall

not attemnpt to derive specific formulas to include all such cases. The set of functions (18) with the Ak not all distinct corresponds to prescription of every U(ak) and of suitable successive derivatives 8"u(ak)/aX" in multiple points ak.

Tf a series of interpolation for harmonic functions is desired, corre-



272 J. L. WALSH..

sponding to interlolation in points some of which are counted multiply, it is necessary to order the points according to increasing multiplicities, so that functions +X,s(z) of increasing subscript correspond to derivatives of higher order at a point.

We have chosen (3) as the basis of our definitioin of norm because of its close relationi to Poisson's integral and the Dirichlet problem and to other classical results. Several generalizations of our entire discussion can be einvisaged: a) use of a positive norm function in (3), b) -use of surface integrals over j z I < 1 in (1) and (3) instead of line iiitegrals, and c) con- sideration of more general regions, not necessarily simply coinnected. The methods that we have used apply also in these cases; the specific formulas are somnewhat different, but at least in the simpler cases can be obtained by the reader.

HARVARD UNIVERSITY.

REFERENCES.

[11 J. L. Walsh, " Interpolation and approximation," Colloquium Publications of the American Mathematical Society, vol. XX, New York, 1935.

[2] 0. Lokki, " tber analytische Funktionen, deren Dirichlet-integral endlich ist und die in gegebenen Punkten vorgeschriebene Werte annehmen," Annates Academaie Scio1tiarum Fennice, A139 (1947).

[3] S. Bergman, ' The kernel function and conformal mapping" Mathematical Surveys, vol. V (1950), New York.

[4] J. L. Walsh and Philip Davis, " Interpolation and orthonormal systems," Journal d'Analyse Mathimatique, vol. 2 (1952), pp. 1-28.



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An Interpolation Problem for Harmonic Functions