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 {|z0z1⋯zn−1|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.
