Arch. Math., Vol. 57, 61-70 (1991) 0003-889X/91/5701-0061 $ 3.50/0 1991 Birkh~iuser Verlag, Basel Formal power series and nearly analytic functions By H~.KAN HEDENMALM 0. Introduction. Consider a formal power series (0.1) Z a.z", n=0 with complex coefficients a.. If (0.2) Ia,I _-< C" for all n for some constant C, then (0.1) is convergent for 1zl < 1/C, and it is analytic there. Conversely, given a function f analytic on {z ~ ~: Izl < r}, its power series representation f(z) = Z a.z" n=O converges for Izl < r, and we have la, I < R" for all sufficiently large n, provided that oo R > r. It seems natural to ask whether we can associate to a given power series Z a, z" 0 a function f having that power series expansion around z = 0, also when (0.2) is violated, in which case the power series diverges unless z = 0. Certainly, this f cannot be analytic near z = 0, because its power series would then satisfy (0.2). It turns out that we can expect f to be nearly analytic, in the sense that Of/8~ be small near z = 0. This will be made more precise in the following section; in particular, the degree of smallness of ~f/8~ around z = 0 is related to the growth of the coefficients {a,}~. A theorem of E. Borel [1] states that to any given formal power series ~ a, x", there is a C ~ function f on the real 0 line such that f(")(O)/n! = a, for all n. The extensions obtained here need not be C ~, but the technique easily modifies to provide C ~o extensions f to the unit disc, and allows us to control [~f] near the origin. Extensions need not be unique. In Section 2 we study how far two functions represent- ing the same divergent power series expansion can deviate from one another. 1. Nearly analytic functions. In the sequel, we shall use the notation 8 = O/Sz and J= 0/0~. Let K) denote the open unit disc in the complex plane IE. Suppose ~o: [0, 1] ~ [0, ~[ is a continuous increasing function. Extend ~o to E) by defining 62 H. HEDENMALM ARCH. MATH. co (z) = co (I z I), z e ID. Introduce the space ~ (co) consisting of all functions f e C (D) with 8f e C (D) and IJf (z)[ _-< CI co(z), z e D, where C$ is a constant depending on f. Endowed with the norm [I f II~(o,) = max { II f II co, II gf/co II ~ }, (co) is a Banach space, and supplied with pointwise multiplication on ID, it is a Banach algebra. Here, [['[f o~ denotes the supremum norm on the region D\{0}; we also used the convention 0/0 = 0. In what follows, we shall assume that co decreases faster than poly- nomially, that is, (1.1) co(r)=O(r") as r~0 for every positive integer n. For 0 < r = 1, let Dr = {zer Izl < r}, and orient its boundary OD, counterclockwise. Let f e ~ (co). By the Cauchy integral formula, we have (1.2) f(z):(2~i)-i t3! f~ (-~-)Zd~"~-~D!~ /kd , z~U r For n = 0, 1, 2 ..... let (1.3) a,(f)--(2rci,-i{~ (-"-lf(()d(+~(-"-i~f(()d(^d(~; the right hand side is well defined because of our assumption (1.1). Also, it is easy to check that a. is independent of the parameter r. We shall see that ~ a. (f)z" is the formal power 0 series expansion associated with f, in the appropriate sense. Proposition 1.1. Suppose f ~ ~ (co), and the a, (f) are given by (1.3). Then (1.4) f (z) = ao(f) + al (f)z + "'" + a._ l (f)z "- I + zn hn(z), where In particular, f (z) = ao(f) + al (f)z + "'" + a,_l (f)z "-I + O(Izl% as z -~ O. P r o o f. We will prove the assertion by induction. When n = 0, (1.4) coincides with the true statement (1.2). Secondly, assuming (1.4) holds for n = k, we will show that it holds for n = k + 1 as well. By (1.3), the definition of h,, and the equality ~-k(~ _ z)-i _ ~-k-1 : z ~-k-, (~ _ z)-~, Vol. 57, 1991 Formal power series and nearly analytic functions 63 we obtain Jf( ) hk(Z)- ak(f)= z(2~i) -1 ~ f f(~) d~ + ~o! ~k+~((_ z) r - z) .J that is, z ~Dr~ hk(Z)-ak(f)= zhk+l(z), z~Dr. It follows that zkhk(Z) = ak(f)z k q- Z k+l hk+ 1 (Z), Z G D,, and the proof is complete. Introduce the weight W[co]: N = {0, 1, 2 .... } ~ [0, oo[ by the relation W[co] (n) = sup {co(t)/tn: 0 < t <= I). Observe that W [co] is an increasing function on N, and that W [co] (n) > 0 for all n unless co = 0. From (1.3) we get the estimate 1 [a,(f)t < [If II| + I[Of/colloo ~t-"co(t)dt 0 by putting r = 1. It follows that ta,(f)l < [If II~)(1 + W[co] (n)). Let's write this as a proposition. Proposition 1.2. Suppose f ~ ~ (co) and that a n (f) is given by (1.3). Then ]a,(f)l < Ilf + W[co] (n)). If we are given an increasing weight sequence w: N ~ [0, oo[, we can construct an associated weight s [w] : [0, 1] ~ [0, oo[ via the relation O [w] (t) = inf {t" w (n) : n ~ N}. The following result explains the duality between the operations W [-] and f2[-]. Proposition 1.3. Suppose co: [0, 1] ~ [0, oo[ and w:N ~ [0, oo[ are continuous increas- ing functions, and that co (0) = O. Then W[f2 [W [co]]] = W[co], and f2 [W[f~ [w]]] = (2 [w]. We have that W [O [w]] = w if and only if log w = {log w (n)}~~ is a convex sequence. In fact, in general, log W[f2[w]] is the largest increasing convex minorant to log w. Moreover, O[W[co]] = co/f and only if the function h(t)=logco(et), t <=O, is concave, and h' (t) ~ IN whenever h' (t) is defined. In general, the function H(t) = logf2[W[co]](et), t < O, is the smallest concave majorant to h with the property that H' (t) ~ N whenever H' (t) is defined. 64 H. HEDENMALM ARCH. MATH. P r o o f. This result is probably well-known in convexity theory. But for the sake of completeness, we include a proof. Let us first show that log W [/2 [w]] is the largest convex minorant to log w. Observe that by a change of variables, log W [f2 [w]] (n) = sup {inf {n - m) s + log w (m): m e N} : s > 0}. In other words, log W [f2 [w]] is the upper envelope of all the lines (1.5) n w-~ inf {(n - m) s + log w (m) : m ~ N} s > 0. The line (1.5) is precisely the line with slope s, lying below the graph of log w, which is closet to the graph of log w. It follows that the upper envelope of the lines (1.5) coincides with the largest increasing convex minorant to log w. We proceed to show that H is the smallest concave majorant to h with the property that H'(t)e N whenever H' (t) is defined. Observe that H(t) = inf {sup {n(t - s) + h(s): s _-< O} : n ~ N} ; in other words, H is the lower envelope of all the lines (1.6) t~sup{n(t-s)+h(s):s=<0}, heN. The line (1.6) is precisely the line with slope n, lying above the graph of h, which is closest to the graph of h. It follows that H, being the lower envelope of the lines (1.6), coincides with the smallest concave majorant to h with piecewise constant N-valued derivative. To show that W[O[W[co]]] = W[co] and O[W[t2[w]]] = f2[w], one need only verify that the pertinent conditions are met by the functions W [co] and f2 [w]. This is left to the interested reader. The following theorem is a partial converse to Proposition 1.2. Theorem 1.4. Let co: [0, 1] ~ [0, oo[ be an increasing continuous function, which has co(l) > 0 and satisfies (1.1). Suppose {a,}~~ is a complex-valued sequence such that (1.7) [a,[ < CW[co] (n - 1)/(n + 1)% n = 1, 2, 3 ..... for some constants C and ct, ct > 2. Then there exists a function f ~ ~ (co) such that a,(f)=a,, hEN. P r o o f. Let g be a continuous function on the unit circle "IF = 01D, to be specified later, and set f(z)- ~ ! ico~g(e:)rdrdO'ore' -z z~lD. Then, by computation, f ~ C (lD), and Jf (r e i~ = co (r) g (egO), so that f ~ ~ (co). From (1.2) and the definition of f, we obtain f(~) ~ ~-~ d~ = 0, zeD, Vol. 57, 1991 Formal power series and nearly analytic functions 65 and by differentiating this equality, we obtain S ;-"-lf(Od( = O, neN. T We conclude that a,(f) = (2hi) -1 SS ~-.-1 gf(Od~ ^ d~ D = -~ ! r-"eo(r)ar. ! e -it"+ 1,Og(eiO)dO 1 = 20(n + 1) S r-"co(r)dr, 0 where 2~ 0 (k) = S e-ikO g (e ~~ dO/2 n.