{an) is said to be admissible for the functionals { Tn I and the class C. The

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1 288 MATHEMATICS: R. C. BUCK PROC. N. A. S. INTERPOLA TION A ND UNIQUENESS OF ENTIRE FUNCTIONS By R. CREIGHTON BUCK* HARVARD UNIVERSITY, CAMBRIDGE, MASS. Communicated July 1, 1947 Let {T. I be a sequence of linear functionals defined on the class K of entire functions of exponential type. We raise the following questions: (I) Given a sequence of mplex numbers I an I and a subclass C of K, is it possible to find a function f belonging to C such that for all n, Tn(f) = a.? (II) Given a subclass C of K and a function f belonging to C, what properties of f can be inferred from the sequence of mplex numbers I Tn(f) I? (III) For what subclass C of K is it true that if f belongs to C and Tn(f) = for all n, then J(z) =? The third is 'of urse a special case of the send, but it is of such special importance that we choose to state it explicitly. A class C for which it holds will be called a uniqueness class for { Tn}. The first question asks for a solution to the general interpolation problem for the class C and the functionals I Tn I; if a function exists having these properties, the sequence {an) is said to be admissible for the functionals { Tn I and the class C. The importance of (III) stems from the fact that if, in (I), C is a subclass of such a uniqueness class then there can exist at most one such function f for which Tn(f) = a., while if, in (II), C is a subclass of a uniqueness class, f is in fact mpletely determined by the sequence { Tn(f) I and every property of f should be inferable from it. This leads naturally to an additional question. (IV) When is it possible to expand f(z) into a series of the form f(z) = Eu.(z) Tn(f)? This is the problem of the existence of an interpolation series for the functionals Tn I, together with the determination of the class of functions for which the series nverges. Clearly, any such expansion class is a subclass of the rresponding uniqueness class. In a previous paper,' certain aspects of these problems were discussed. Here, the methods used there-due essentially to P6lya. and Carlson3-are given in a more general form, and are applied to the solution of problem (IV).' The treatment is similar to that of Gelfond,4 but is of much greater power;' most of the known results on nvergence of Abel, Newton, Stirling, Selberg, as well as other less familiar interpolation series, are obtained at once, and the method extends to discussion of summability. In the present paper, the general method is outlined briefly, and a few of the specific results are stated; detailed proofs will appear in a later paper.

2 ttol. 33, 1947 VLA THEMA TICS: R. C. BUCK 289) We first seek general representations for functions f and functionals T. Some preliminary definitions are necessary. If f e K, the class of entire functions of exponential type, so that there exist real numbers A and C such that lf(z)l <. AeCIS`, then the growth function h(o, f) defined by lim sup r-1 log Jf(re"6)I is bounded in absolute value by C. If H(O) is any function with period 27r, then K(H(O)) denotes the class of allf in K such that h(o, f) < H(O) for all ; H() may take infinite values. As in previous papers, the notation K(a, c) is reserved for the class of all f in K with h(o,f) < a, h(7r, f) K a, and h( 7, f) % c. If G is any closed set o the 2' mplex plane, k(, G) sup tr{wei'} is called the supporting function of w eg G; this differs from the usual definition in the sign of. The function k(, G) is ntmuous in, has period 27r, has left and right derivatives everywhere, and is unchanged if G is replaced by its nvex hull. If G1 and G2 are point sets, then G1 G2 will be the set of all points of the form zz' where z' e G1, z' e G2.5 If G1 and G2 are bounded so is G1*G2; if G1 and G2are closed and mpact, so is G1 G2; if G1 and G2 are closed bounded nvex sets, then GI G2 is simply nnected; GC1 G2 is a star set whenever one - factor is a star set. If f(z) = Ea.zn/n!, belonging to K, then +(w) = a E an/wn+l is usually called the Borel tratnsform of f. We denote by D(f) the nvex hull of the singularities of +(w). In our notation, a fundamental theorem due to Polya2 states that if f belongs to K then hi(, f) = k(, D(f)) for all. Suppose we nsider a function y (z) = EZn/n! c. and an associated func ~~~~~~~~~~ tion y*(z) = Zcnz/n/!, both in K. A modified Borel transform of f(z) is X +(w) is regular outside the set D(f).D(y*), defined by +(w) = EanCcn/wn+1. and if r is any ntotir enclosing this set, f(z) has the representation f(z) = r fr y(zw)k(w)dw. (1) Conversely, if y e K, if G is a closed bounded simply nnected set, if +(w) is regular outside G and if r is a ntour enclosing G and not passing through the origin, then f(z), defined by (1), belongs to K, and h(, f) ( k(, G'D(J)). (The special case for which y(z) =e = -y*(z) was studied by P61ya; it is of interest that if D(zy) = D(y*) = 1, then y(z) = Ae'.) Turning now to functionals, we may obtain a wide variety as follows. Let +(w) be a modified Borel transform of f with kernel y(z), and choose any ntour r enclosing D(f) DD(y*). Then, for any entire function g(w), we define T by:

3 29 MA THEMA TICS: R. C. B UCK PROC. N. A. S. T(f) = y fr g(w)4o(w)dw. (2) This includes all of the usual cases. In fact, if K is given the weak topology, any ntinuous linear functional on K may be represented in the form (2). When T is so defined, g(w) is said to be the generating function of T. With this background, we may now study general interpolation series. For simplicity, we nfine ourselves to the case where the functionals TnJ have generating functions {gn(w)} which are exactly the integral powers of a function P(w). This restriction is indeed satisfied for most of the familiar series. If D(w) is regular and univalent in an open set Qw of the w-plane, ntaining the origin, and if Q2w rresponds to a set Or in the D- plane, then w = w(r), regular in Or. Since y(zw) = y(zw(r)) is analytic in R for D in QR,,y(zw) = E u,(z) n (3) uniformly nvergent in any closed subset of Ar, the largest open circle j < R ntained in Or. Let A. be the image of Ar, given by r(w)i < R. Then y (zw) = u,(z)[r(w)] n o (4) = EI Un(Z)gn(W) is uniformly nvergent in any closed subset of A.. If f is such that D(f)- D(Qy*) is interior to A., then we can choose a ntour r lying in A,, and enclosing D(f) -D( y*), on which (4) is uniformly nvergent. Applying (1) and (2), and integrating termwise, we obtain the nvergent interpolation series f(z) = un(z)tn(f). (5) The same procedure will yield results for summability. Let E(t) = E dntn be an entire function, not a polynomial, with dn. If {S.} is a. sequence of mplex numbers with S,ll"" = (1) then E Sndntn nverges for all t to a function H(t). The limit lim H(t)/E(t), if it exists, is the I-* +a generalized E-limit of {Sn Applied to a power series, the following is true: if f(z) is regular in a region R ntaining the origin, then

4 VoL. 33, 1947 MA THEMA TICS: R. C. B UCK 291 f(z) = E - E f(n) () zn/n! (6) where the summability is uniform in any closed mpact subset of (R'- E')'. Here, E is an open set such that lim E(zt)/E(t) is zero, uniformly in any mpact subset of E. Returning now to our discussion of interpolation series, we take up E- summability. Since 'y(zw), as a function of ~, is regular in Q4, we have (zw) = E - E 8(Z) n uniformly in any mpact subset of QR = (r'-e')'. - If Ow is the image of Q* under w = w(r), then y(zw) = E - E Un(Z)gn(W) uniformly in any mpact subset of Q. Proceeding as before, we substitute this into (1), and integrate termwise; simplifying by (2), we have f(z) = E - un(z) Tn(f), (7) holding for all f such that D(f) D(y*) lies interior to fg. We summarize these results as follows. THEOREM. Let gn(w) = [v(w)] n be the generating functions of the functionals {Tn. Then, the formal interpolation series E un(z) Tn(f) nverges to f(z) for all f such that D(f) DQ(y*) lies in Aw, and is E summable to f(z) for all f such that D(f).D(y*) lies in Q*. By specialization of the kernel yy(z), and the function (w) and the function E(t) determining the method of summability, many specific results, both new and old, can be obtained. We indicate briefly only a few of these; more detailed treatments will appear later. THEOREM 1. The Newton expansion z() Anf(O) is nvergent to f(z) if h(o, f) < sin + s log (2 s ), is Borel summable to f(z) iff e K(a, c) with c < and is Mittag-leffler summable to f(z) iff e K(a, c), with c < r. COROLLARY (Carlson6). Iff e K(a, c), c < r, and f(n) = Ofor n =, 1, 2,..., then f(z) =. THEOREM 2. The Abel series, z(z - n)n-1 f(n)(n)/n! is nvergent to f(z) if D(f) lies in the region wel+wl < 1, and is ML summable to f(z) if

5 292 MA THEMA TICS: R. C. B UCK PROC. N. A.- S h(o, f) < k(o, Q ) where Sl* is the region bounded by the curve p (ir - I<pj)lsin jo. COROLLARY. Iff e K(A), A < 1, andf(n)(n) = Oforn =,1,...,then f(z) =. -1 THEOREM 3. The S&irling series f() + Z ( n+712) Anf(-:) is nvergent to f(z) if D(f) lies in the region Isink (w/2)1 < 1, is Borel summable to f(z) if f e K(r/ /2), and is ML summable if f e K(a, c) with c<ir. THEOREM 4. The series f(o) + z Z n ( n 1) Anf(n) is nvergent tof(z) iff ek(a), with A <log (1 + N/72)/2andisMLsummableif h(o,f)< (-) sino+so log (-2sO) for 2 Ioj r. THEOREM 5. If D(f) lies in the region bounded by the curve u = log sin,v- log sin (, + 1)v, for >, then f(z) = ML- {f(o) + f z n( on 1 ) (fofgn)}. Several of the above theorems can be strengthened nsiderably, but for simplicity of statement only the weaker forms have been given. In particular, the uniqueness theorems obtained by Gelfond4 by methods of nformal mapping beme immediate rollaries of theorems on the summability of the rresponding interpolation series. It is also possible to treat series arising from sequences of functionals { Tn I -whose generating functions are not of the form [v(w)]_. The Lidstone series7 is a simple example of such a series. THEOREM 6. The Lidstone series EI An(z)f (2n) (1) + E An( - z) f (2 )() are nvergent to f(z) if f e K(A), A < r, are Borel summable to f(z) if f e K(a, c), c < xr, and are ML summable to f(z) if D(f) does not ntain any points of the vertical lines u =, Iv K 7r. * Society of Fellows, Harvard University. 1 Buck, R. C., Duke Journal, 13, (1946). 2 P6lya, G., Math. Zeit., 29, (1929). 3 Carlson, F., Sur une classe de series de Taylor, Thesis, Uppsala. I Gelfond, A., Math. Sbornik, 4 (new series), (1938X. 6 Not to be nfused with G, n G2 which denotes the intersection of the sets G1 and G,. In general, G' will denote the mplement of G. 6 Hardy, G. H., Acta Math., 42, (192). 7 Boas, R. P., Jr., Duke Journal, 1, (1943).