Sampling expansions for functions having values in a Banach space

Abstract

A sampling expansion for vector-valued functions having values in a Banach space, together with an inversion formula, is derived. The proof uses the concept of framing models of Banach spaces that generalizes the notion of frames in Hilbert spaces. Two examples illustrating the results are given, one involving functions having values in L p [ − π , π ] , 1 > p ≤ 2 L^{p}[-\pi , \pi ], 1>p\leq 2 , and the second involving functions having values in L p ( R ) L^{p}(\mathbb {R}) for 1 > p > ∞ . 1 > p> \infty .