Appell polynomial expansions and biorthogonal expansions in Banach spaces

Abstract

Let { p k } 0 ∞ \{ {p_k}\} _0^\infty denote the sequence of Appell polynomials generated by an analytic function ϕ \phi with the property that the power series for θ = 1 / ϕ \theta = 1/\phi has a larger radius of convergence than the power series for ϕ \phi . The expansion and uniqueness properties of { p k } \{ {p_k}\} are determined completely. In particular, it is shown that the only convergent { p k } \{ {p_k}\} expansions are basic series, and that there are no nontrivial representations of 0. An underlying Banach space structure of these expansions is also studied.