<Emphasis Type="Italic">q</Emphasis>-Besselian Frames in Banach Spaces

In this paper, we introduce the concepts of q-Besselian frame and (p, σ)-near Riesz basis in a Banach space, where σ is a finite subset of positive integers and 1/p + 1/q = 1 with p > 1, q > 1, and determine the relations among q-frame, p-Riesz basis, q-Besselian frame and (p, σ)-near Riesz basis in a Banach space. We also give some sufficient and necessary conditions on a q-Besselian frame for a Banach space. In particular, we prove reconstruction formulas for Banach spaces X and X* that if is a q-Besselian frame for X, then there exists a p-Besselian frame for X* such that for all x ∈ X, and for all x* ∈ X*. Lastly, we consider the stability of a q-Besselian frame for the Banach space X under perturbation. Some results of J. R. Holub, P. G. Casazza, O. Christensen and others in Hilbert spaces are extended to Banach spaces. This work is supported by the Natural Science Foundation of Fujian Province, China (No. Z0511013) and the Education Commission Foundation of Fujian Province, China (No. JB04038)