  In this paper we characterize the zero sets of functions from $\ell^{p}_{A}$
(the analytic functions on the open unit disk $D$ whose Taylor coefficients
form an $\ell^p$ sequence) by developing a concept of an "inner function"
modeled by Beurling's discussion of the Hilbert space $\ell^{2}_{A}$ (the
classical Hardy space). The zero set criterion is used to construct families of
zero sets which are not covered by classical results. In particular, it is
proved that when $p > 2$, there are zero sets for $\ell^{p}_{A}$ which are not
Blaschke sequences.
