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. 0 It follows that }1 O(n + 1) = a,(f) t-"co(t)dt , neN.

If we can find a 0 s C (1") having

(1.8) O(n+l)=a. {i t-"co(t)& }1 , neN,

1 the assertion a. (f) = a. will follow. We will need to estimate S t-" co (t) dt from below. Let 0 t, e ]0, 1] be a point where sup {t -"+ 1 co (t) : 0 < t < 1} is attained, that is, t~""+1 co(t,)= W[co] (n- 1).

Observe that t, ~ fl as n ~ 0% where fl = inf supp co = inf {t e [0, 1] : co (t) > 0}, which has 0 < fl < 1, since we assumed that co (1) > 0. Now since co was increasing,

1 1 1 S t-"co(t) dt ~ S t-"co(t)dt >= co(t.) S t-ndt = co(t.)" (t~-"+l - 1)In. 0 tn tn Since t. ~ fl e [0, 1[, we have for sufficiently large n, (t~-"+l - 1)In > t~"+l/2n, so that 1 (1.9) S t-" co(t)& > t2 .+1 co(t,)/2n = w[co] (n -- 1)/2 n. o By (1.7) and (1.9), we have that for big n,

la.I t-"co(t)d <2C/n ~-1 ,

Archly der Mathematik 57 5 66 H. HEDENMALM ARCH. MATH. so that

la.I t-"~o(t) d < oo. n=0 {i It is now possible to solve the problem (1.8); in fact, we can solve it with g being a disc algebra function with summable Taylor series. The proof is complete. R e m a r k. The technique used in the proof of Theorem 1.4 resembles that used by Dynkin in [2], where he studied classes of almost analytic functions on the unit circle T.

2. Uniqueness of extensions. Suppose two functions F, G ~ ~ (co) have the same formal power series expansions at 0, that is, a.(F) = a,(G) for all n = 0, 1, 2 ..... By Proposi- tion 1.1, this means that the function f -- F - G satisfies (2.1) f(z) = O(Izl") as z ~ 0 for all n = 1, 2, 3 ..... If co (t) = 0 on [0, e] for some e > 0, then f e ~ (co) implies that f is analytic on D(0, e), the open disc with radius e, centered at 0, so by (2.1), (2.2) f--0 on D(0, e). Assertion (2.2) is considerably stronger than (2.1). This leads one to wonder if perhaps in general, provided f e N (co), (2.1) can be strengthened to something like (2.3) l/(z)l < Cso)(z), z~ID, for some constant C; depending on f. Unless co vanishes on some interval [0, ~] with e > 0, we certainly cannot hope to get anything as strong as (2.2). To see this, let {~,}g c] 0, 1[ be a strictly decreasing sequence, converging to 0, and let {f, } ~ a corresponding sequence of smooth nonidentically vanishing functions on E), satisfying f,(z) = 0 if [zl < e, or Iz[ > 8.-1. Given an co which is positive on ]0, 1], it is not difficult to choose positive real numbers 6, so as to make the series

f(z)= ~ 6,f.(z), z~lD, n=l converge in ~ (o~). The function f is an element of N (co), satisfies (2.1), and yet it does not meet (2.2) for any e > 0. Now suppose we have an arbitrary f e N (co) with a. (f) = 0 for n = 0, 1, 2,.... We may assume without loss of generality that f = 0 on T; otherwise we simply replace fby fl=f'z, wherex~C~,0

(2.4) f(z) = z"(2ni) -1 ~-f(~) d~/x d~-, z e lD.

Since I~f(~)/~"l ~ C~ 14"l-"~o(l~l) Vol. 57, 1991 Formal power series and nearly analytic functions 67 for some constant C: depending on f, and

ff dA(~) <27r, zeD, 1~ - zl = where dA is planar area measure, we have the estimate

(2~ i)-~;.~ Jf(O d( ^ d( <2C:sup{r-"'co(r):O

= 2 C:" W[co] (n).

By (2.4), it now follows that

If(z)[ < 2C:" W[co] (n)[z[" for all n = 0, 1, 2 ..... so that

(2.5) If(z)[ < 2C:inf{W[co] (n) Izl": n = 0, 1,2 .... } = 2C:O[W[co]] (Izl). Proposition 1.3 descripes in detail when f2[W[co]] = co. At any rate, the previously mentioned observation (2.2) follows from (2.5) in the case when co = 0 on D (0, 0- It is possible to ref'me estimate (2.5). Let q E N\{0}, and put

(z)=f(z q), z~D. Now 8fq (z) = qSq-1. ~f (z q) from which we deduce that f e ~(coq), where coq(r)=:-lco(rq), O<-r<- l.

One easily checks that coq is a continuous increasing weight which satisfies (1.1) because co does. Since If(z)[ = O (}z]") as z ~ 0 for all n = 1, 2, 3 ..... we have [f~ (z)[ = O ([z[") as z --* 0 for all n = 1, 2, 3 .... as well, and hence a,(fq) = 0 for all n E N. Let k e ]hi. By Proposition 1.1,

fq(z) = z k. (2 rc i) -1. II q(q-~ ~-k(( _ z)-~ Of((~)d~ ^ d~, D from which we derive the estimate

tfq (z)[ _-< q Izl k" t1 r ~ (~ - z) -1 IILI(~) II ( ~ ~-q-1 [k ~-f ([q) IIL~(tD) <= 2 q Cy lz[ k sup {rq-k-l co(rq): O < r <= 1} = 2qC: [z[k sup {t l-(k+l)/q co(t): 0 < t_< 1}

= 2qGlzl k w[col ((k + 1)/q - 1), where C~ is the same constant as last time. If we write p = k + 1 - q, we get

If(z)l < 2q Cy ]zl (p- a)/~+ 1 W[co] (p/q).

5* 68 H. HEDENMALM ARCH. MATH.

The condition k 6 N translates to p + q e Z+ = N\{0}. If we substitute np for p and nq for q, where n EZ+, we get

(2.6) If(z)l<=2qC:lzlm+lW[og](p/q)inf{nlzl-1/c"q):n~Tg+); this is permissible because np + nq ~ 2g+ if p + q ~ 7/.+. We should study the quantity inf {nq Izl- x/~q): n e L}. Calculus tells us to choose n near q-1 log l/lzl. So, let n o be the smallest integer > q-1 log l/Izl; then (2.7) inf{nqlzl-l/~"q):neZ+}

(2.10) If(z)] _-__2eC: [z[ inf {(q + log 1/[zDf2q[W[o9]] (tz]): q ~ 2~+}.

Let us assume ~o is such that the function g(t)=--logco(e-'), t >O, is smooth, convex, and increasing. Let us decide that we want to know when (2.10) implies that If(z)l < K:,o "lzl (1 + log 1/tz[)" e)(z), z E D\{0}, for some constant K:, ~ depending on f and co. Put G a (t) = - H a (- t), where H a is as above. Then G a can be interpreted as the biggest convex minorant to 9 with the property that qG~(t)~ N whenever G~(t) is defined. If we substitute z = e -t+i~ where t _>- 0 and 0 ~ N, (2.10) becomes (2.1t) If(e-'+~~ < 2eC:e-'inf{(q + t)exp(-Ga(t)):qe2~+}. If we take the infimum in (2.11) only over those q e Z+ such that for some fixed constant K > 1, q < qx(t), where qK(t) is the biggest integer < K + (K - 1)t, then (2.11) becomes

(2.12) If(e-'+'~ < 2eKC:(1 + t)e -t exp (-sup {G~(t): Z+ ~ q < qr(t)}). Vol. 57, 1991 Formal power series and nearly analytic functions 69

Our problem becomes to determine for which smooth, convex, and increasing functions g it is true that sup {Gq(t): 2~+ ~ q < q~(t)} > g(t) - M, for some constant M > 0. After a moment's thought we realize that this certainly holds if (i) g' (z) -- g' (t) < 1/q K (t) implies that (ii) g (z) - g (t) - (~ - t)g' (t) < M, for all z _> t >- 0. Let q~: [0, oo [-*]0, oo[ be a continuous function, and let ~ denote an antiderivative to q~, that is, a solution to ~' (t) = ~b (t). Let us assume that g" (t) > ~b (t) for all t > 0. We hope that for some q~, this condition will imply that (i) ~ (ii). So, assume (i) holds, but not (ii); we hope to arrive at a contradiction. Observe that

g' (z) -- g' (t) = ~' O" (s) ds >= i c~ (s) ds = (z) - (t), t t so that (2.13) ~(z) - ~(t) < I/qK(t ). Also, since g is convex,

g(z) -- g(t) -- (z -- t)g' (t) = i (g' (s) -- g' (t))ds <= i (g' (z) - g' (t))ds t t

= (~ - t)(a'(~) - a'(t)), so that since (i) holds and (ii) doesn't, (2.14) z - t >= M . qK(t). We want to find a such that (2.13) and (2.14) are incompatible for all z > t _> 0, that is, we should have (2.15) qb(t + U. qr(t)) - q~(t) >= l/qK(t ) for all t --- 0. It is not difficult to check that the function ~(t)=-8(1 +0 -1 , t=0, satisfies (2.15), provided that I+MK 8>_ - M(K -- 1) 2" If ~ < 1, this inequality holds if we choose M = 1 and K = 4/e. We have obtained the following theorem.

Theorem 2.1. Suppose co is such that the function g(t) = -log~o(e-t), t > 0, 70 H. HEDENMALM ARCH. MATH, is smooth, convex, increasing, and satisfies g"(t)>5(l+t) -2, t>O, for some 5, 0 < 5 < 1. Let f ~ ~ (o9) vanish on T and satisfy a, ( f ) = O for all n = O, 1, 2,...; then If(z)l =< 85 -le 2 Cr Izl (1 + log 1/[zl)~o(z), z ~ D\{0}, where Cj, = I1 ~f/o9 II oo. R e m a r k. If we want a statement for the case when f might not vanish on T, we have to carry though the argument with the cut-off function • quantitatively. After a few computations, one then obtains the estimate

If(z)l < 8(1 + 10/r -1 e z II Jf/coll~ Izl (1 + log 1/Izl)oJ(z), for 0 < Izl < 1/2. References [1] E. BOREL, Sur quelques points de la th~orie des fonctions. Ann. ]~cole Norm. (3) 12, 44 (1895). [2] E. M. DYNKIN, Functions with a prescribed bound for af/a~, and a theorem of N. Levinson. Mat. Sbornik (N.S.) 89 (131) 182-190 (1972); English translation in Math. USSR Sbornik 18, 181-189 (1972).

Eingegangen am24.10.1989 AnschrifldesAutors: H~kan Hedenmalm Department of Mathematics UppsalaUniversity Thunbergsv~gen 3 S-75238Uppsala Sweden