Weighted l('2) approximation of entire functions and related topics

Abstract

In Chapter 1, sufficient conditions for polynomials to be dense in the space of entire functions of· L2 (dm) are examined, where dm is a positive, absolutely continuous measure defined on the complex plane. Let S be the space of entire functions such that [] Write dm(z) as K(z)dxdym K(r,0)rdrd6. The main theorems are: 1) Suppose In inf f K(r,8) is asymptotic to in s6up K(r,6) (together with other mild restrictions). Then polynomials are dense in S. 2) Let K(z) = e-$(z) where ♦(z) is a convex function of z such that et z belongs to S for all complex t. Then the exponentials are complete ins. (Corollary: Polynomials are dense ins.) l) is extended to the several-variable case. 2) was recently proven by B. A. Taylor for the many-variable situation. our proof does not extend beyond the case of one variable, but for this it is simpler and more direct than Taylor's. Examples of spaces in which polynomials are not dense are also given. (from Introduction)