Vol. 44, No. 1, 1973

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On the behavior of Pincherle basis functions

Maynard G. Arsove

Vol. 44 (1973), No. 1, 13–31

A basis {αn} in the space of analytic functions on a disc {z : |z| < R} is called a Pincherle basis if, for each n(= 0,1,), the Taylor expansion of αn(z) has zn as its first nonvanishing term. The object of the present work is to examine such sequences to determine how behavior of the individual functions αn is related to the property that {αn} is a basis. Of particular interest are the zeros of the functions ψn(z) = αn(z)∕zn, and the case when each ψn is a linear function vanishing at a corresponding point zn is studied in detail. There exist bases in which infinitely many of the zn coincide with some point in the disc, or in which the zn cluster at the origin. Nevertheless, the basis property can be correlated with various growth rate conditions on {zn}. For example, if the sequence {|z0z1zn1|1∕n} converges to some number A, then the condition A R is necessary and sufficient for {αn} 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) ψn, and certain phenomena encountered in this case are illustrated by examples involving nowhere vanishing ψn.

Mathematical Subject Classification
Primary: 30A98
Received: 24 July 1971
Published: 1 January 1973
Maynard G. Arsove