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Mean Convergence of Vector‐valued Walsh Series
Author(s) -
Wenzel Jörg
Publication year - 1993
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19931620109
Subject(s) - series (stratigraphy) , convergence (economics) , mathematics , citation , combinatorics , arithmetic , calculus (dental) , algebra over a field , computer science , library science , pure mathematics , medicine , paleontology , dentistry , economics , biology , economic growth
Given any Banach space $X$, let $L_2^X$ denote the Banach space of allmeasurable functions $f:[0,1]\to X$ for which ||f||_2:=(int_0^1 ||f(t)||^2 dt)^{1/2} is finite. We show that $X$ is a UMD--space (see \cite{BUR:1986}) if andonly if \lim_n||f-S_n(f)||_2=0 for all $f\in L_2^X$, 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 $(w_i)$.