REMAINDERS OF POWER SERIES

Suppose ∑n=0∞anzn has radius of convergence R and σN(z)=|∑n=N∞anzn|. Suppose |z1|<|z2|<R, and T is either z2 or a neighborhood of z2. Put S={N|σN(z1)>σN(z) for zϵT}. Two questions are asked: (a) can S be cofinite? (b) can S be infinite? This paper provides some answers to these questions. The answer to (a) is no, even if T=z2. The answer to (b) is no, for T=z2 if liman=a≠0. Examples show (b) is possible if T=z2 and for T a neighborhood of z2.

KEY WORDS AND PHRASES.Power-Series, Remainders, Radius of Convergence.AMS (MOS) SUBJECT CLASSIFICATION (1970)  CODES.30AI0.1. INTRODUCTION.This paper originated in a question of approximation by power series raised in Query 51 in the American Mathematical Society Notices Ill. (The query originated in considerations of analytically continuing a polynomial series from the interval [-1,1] to the region of convergence of the series.)Suppose f(z) m n m n n=O anZ has radius of convergence R and ON(Z) IXn=N anZ I. Suppose Zll < z21 < R and T is either z 2 or a neighborhood of z 2. Put S {nlOn(ZI) > On(Z) for z g T}. S is cofinite if its complement is finite.Two questions are asked-(a) can S be cofinite?(b) can S be infinite?
One might expect the answer to both questions to be no since one expects the approximation to f by partial sums of its power series to be worse, closer to the circle of convergence.
This paper provides some answers to these questions.Section 2 shows (a) is impossible for any T. Section 3 shows (b) is impossible if T z 2 and lim an a # 0. Section 4 shows (b) is possible for T z 2 and Section 5 shows (b) is possible for T a neighborhood of z 2.
Section 5 suggests the conjecture that if T is a neighborhood of z2, then S must be "thin."The S which appears in Section 5 is lacunary.
These questions can also be raised about other series of orthonormal polynomials with elliptic domains of convergence.(cf.Szeg [5], pp. 309-10).
2. S CANNOT BE COFINITE.
The following theorem was suggested by P. Lax [3].PROOF.Suppose S contains a nonempty tail set I; i.e. n_l implies n+l I. Then for n, Suppose 1/R 0. Choose g > 0 so that (R -1 + e)[Zl[ < 1 and choose n z so large that lakl 1/k < (1/R + ) for k _> n.Also choose n so, that lan I1/n > I/R e,. Then Now in addition to the other conditions on n, choose n large enough so that Then, since Iz2l Now choose n.x so that (61Zll/Iz21)ni < 5 -1 Then ei IZll Iz21 (1-e ilzl) The following observation about general series was made by a referee.Let O Ap be convergent If 0 p lbpl < then

CASE OF LIMN A
In this section it is shown that (b) is impossible for even a single point if limn an a # 0. The proof is as follows.For > 0, N large enough, The following example shows (b and thus O2k(I/2) 0. So for any z I 1/2 and 0 < Izll < I, O2k(Z I) > O2k(1/2).
Note that for an e-neighborhood of 1/2" N {zl Iz 1/21 < 0 < g < I/2 and for any z I with ]z I] < I/2 g, O2k(Z I) converges to zero faster than (2k(Z) at any point z in N except I/2.So we cannot extend the result to a neighborhood of 1/2. 5. CASE OF T A NEIGHBORHOOD OF z 2. THEOREM 2. For each R, 0 < R _< m, there exist points z and z 2 with IZll < Iz21 < R and a power series 2m n n=0 anZ with radius of convergence R such that for infinitely many values of N, ON(Zl)/3 ON(Z) for all z in some neigh- borhood of z 2.
PROOF.Suppose R I. Put n k 4 k and Pk(Z) (]/b k) zn2k-I (z I/2)n2k, Note that and n2k + n2k-1 < n2k+l (2) n2k_l (log 4/log 3 + I) < n2k (3) for all k.(2) implies that each a is either zero or appears exactly once as n a coefficient in the expansion of some Pk(Z).Let Jk be the integer for which max0<j<n2k v\ Then aj+n2k_l 1/(j +n2k This is less than or equal to one for all j and equal to one for j Jk' which implies the radius of convergence is one. For all z with z 1/21 < 1/4" GN(r) exp -o log oN(re10)d0 (r < R) and the p th mean, p > O- o(rei0)d 0 (r < R) are both monotone increasing functions of r for each N and log GN(r)and log IN(r) are convex functions of log r.Thus in the geometric mean sense and pth P mean sense, oN(z) become larger as one approaches the circle of convergence.
of convergence R. It follows from results n in Plya and Szeg [4, Part III, problems 307-310] that the geometric mean"