On Uniqueness Sets for Expansions in Sequences of Functions Arising from Singular Generating Functions

Abstract

Let {pn(z)}; be a sequence of functions analytic in a region D. A problem in analysis which has received much attention is the following: describe the sets Z ⊂ D for which(1)implies hn is 0 for all n, (To make the problem interesting, only those situations are studied where finite subsets of the pn(z) are linearly independent in D.) Another way of phrasing this is: Characterize the uniqueness sets of pn(z), a uniqueness set Z being a set in D such that the restriction of {pn(z)}; to Z is linearly independent. If Z is not a uniqueness set then for some {hn}; not all 0, we have(2)This formula is called a non-trivial representation of 0 (on Z).