New York Journal of Mathematics
Volume 21 (2015) 973-986


Earl Berkson

Multipliers in weighted settings and strong convergence of associated operator-valued Fourier series

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Published: September 27, 2015
Keywords: Ap weight sequence, shift operators, Fourier multiplier
Subject: Primary 42A20, 42A45, 46E30

This note describes the pleasant features that accrue in weighted settings when the partial sums of the operator-valued Fourier series corresponding to a multiplier function ψ :TC are uniformly bounded in operator norm. This circle of ideas also includes a Tauberian-type condition on the multiplier function ψ sufficient to insure such uniform boundedness of partial sums. These considerations are shown to apply to Riemann's continuous, "sparsely differentiable,'' periodic function. In a larger sense, our considerations aim at showing how pillars of functional analysis and real-varable methods in Fourier analysis can be combined with "bread-and-butter'' techniques from these subjects so as to reveal hitherto unnoticed useful tools in multiplier theory for weighted Lebesgue spaces.

Author information

Department of Mathematics; University of Illinois; 1409 W. Green Street; Urbana, IL 61801 USA