A Universal Power Series for Approximation of Measurable Functions
We prove the existence of "universally overconvergent" power series. I.e., i f O g R c o o , then there exists a power series P(z) which converges absolutely for |z| g R such that to every complex-valued measurable function /(z) there exists a subsequence of partial sums of P(z) converging to/(z) a. e. for |z| > R. In fact, the set of all such universally overconvergent power series is dense in the Banach space of all functions which are continuous on and analytic on the interior of the disc |z| furnished with the supremum norm. AMS-classification: 28-A20, 40-A30, 41-A10. 1. Statement of the results Consider a fixed real number 0 ^ R < o o and denote by a)JTfl= {Kn} a sequence of sets /C„ c {|z| >/?} such that K I S K 2 S . . . , (1) Kn is connected, K„ compact for η e Ν (2) b) Jf (jTr) the set of all functions / :  R such that for every sequence Jf R = {£„}„!?! and every feJCiJfg) there exists a subsequence n} / o o such that for Sn{z) 2 L. Tomm and R. Trautner "j = Σ akz the statements k = 0 S n ( z ) ^ ^ f ( z ) in \z\>R (5) Snj(z)^f(z)foralln= 1 ,2 , . . . (6) hold. The power series Ρ (ζ) in question is a "universal power series" for i) almost everywhere convergence in |z| > R ii) uniform convergence on every compact set Κ<= {z:|z| > R } where K is connected, iii) compact convergence on every open set G c= \z:\z\>R\ the components of which 00 are simply connected (note that any such G can be exhausted by K„, G= {jK„ for ι suitable X'R = {£„}, such that any compact K c G i s contained in a Kn). Clearly the limit function which can occur in i-iii must be measurable on Izl > R , ο Ο continuous on Κ and analytic on K, analytic on G respectively. This shows that the most general functions which may be admitted can be approximated in the suitable sense by the partial sums of P(z). Note that the power series P(z) does not depend on the class J f^ = {/£„}„=, but is universal for all such We remark that a slightly more general version of the theorem may be obtained by admitting A. that/(z) assumes the value oo on a certain subset of {|z| > R}, if almost everywhere and uniform convergence to oo is interpreted suitably; B. that snj(z) should by bounded, but almost everywhere divergent on a prescribed 00 measurable subset Μ c {|z| > R}\ LJ Kn n= ι The theorem stated above covers and generalizes results of Chui-Parnes [1], Luh [2] and Luh-Trautner [3]. The basic tool for its proof is the following supplement to Mergelyan's approximation theorem recently obtained by Tomm [4], Theorem A. Let be given a) any sequence JT = {K„}„®, satisfying (1, 2), b) any function/(z) satisfying (3, 4). Then there exists a sequence of polynomials \Qf z)) with rational complex* coefficients such that β;(ζ)—/(z) a. e. in C , (7) Qj(z)->f(z) uniformly on Kv for v = l , 2 , . . . . (8) 2. Proof of the Theorem Consider the set Γ of all polygons which run from the point R through a finite number of rational complex vertices to a rational complex number and from there radially to * By a rational complex number we understand a number of the form a + b-i {a, feeQ). A Universal Power Series for Approximation of Measurable Functions 3 infinity. Also consider the set Q of all polynomials with rational complex coefficients. Clearly Γ and Q and therefore Γ xQ are countable sets, e. g. Γχ<2 = {(γ„, qn)i7„er, qn e ß , η ε Ν } . For n = 1, 2 , . . . we can find connected open sets Un such that y . s l / . c U { t : | z t | < i (9) zey„ (. n j μ(ϋη)< 2 " " , ( 1 0 ) We define for η = 1, 2, . . . W„ = Unn{z :R+-^\z\^R + n}. ( 1 1 ) η So the Wn are compact, their complements are connected. We now construct the power 00 series P(z)= Σ akz where the coefficients ak are determined by the following ink = ο ductive definition of the partial sums vn Σ a k z k : k = 0