Mean Convergence of Vector--valued Walsh Series

1992/10/30 by Wenzel, Joerg
#46E #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.math/9210208

Abstract

Given any Banach space X, let L₂^X denote the Banach space of all measurable functions f:[0,1]→ X for which ||f||₂:=(int₀¹ ||f(t)||² dt)^1/2 is finite. We show that X is a UMD--space (see \citeBUR:1986) if and only if lim_n||f-S_n(f)||₂=0 for all f∈ L₂^X, where S_n(f):=sum_i=0^n-1 (f,w_i)w_i is the n--th partial sum associated with the Walsh system (w_i).

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