On the behavior of Pincherle basis functions
Maynard G. Arsove · Pacific Journal of Mathematics · 1973
A basis {a n } in the space of analytic functions on a disc {z: I z I < R} is called a Pincherle basis if, for each n(=0,1, )» the Taylor expansion of a n (z) has z n as its first nonvanishing term.The object of the present work is to examine such sequences to determine how behavior of the individual functions a n is related to the property that {a n } is a basis.Of particular interest are the zeros of the functions f n (z) = a n (z)lz n 9 and the case when each f n is a linear function vanishing at a corresponding point z n is studied in detail.There exist bases in which infinitely many of the z n coincide with some point in the disc, or in which the z n cluster at the origin.Nevertheless, the basis property can be correlated with various growth-rate conditions on {z n }.For example, if the sequence {| z o zι Zn-ι \ l/n } converges to some number A, then the condition A ^ R is necessary and sufficient for {a n } to be a basis.This and similar results are derived by using the automorphism theorem and properties of entire functions of exponential type.Correlations of this sort fail to materialize, however, for general (nonlinear) f n , and certain phenomena encountered in this case are illustrated by examples involving nowhere vanishing f n .