Convergence of Sequences of Convolution Operators

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New Zealand Journal of Mathematics

Vol. 38, (2008), Pages 137-147

J. M. Rosenblatt

Department of Mathematics

University of Illinois at Urbana

Urbana, IL 61801

United States of America

mailto:rosnbltt@illinois.edu

Abstract When $(φ n)$ is a sequence of positive functions in $L_1(\mathbb R)$ such that the convolutions $\phi_n\star f$ converge in $L 1$ -norm to $f$ for all $f \in L_1(\mathbb R)$ , then these convolutions may or may not converge almost everywhere too. There is always a subsequence $(\phi_{n_k})$ such that at least $\phi_{n_k} \star f$ converges almost everywhere to $f$ for all $f \in L_p(\mathbb R), 1 < p < \infty$ . A special case of this is when the functions $φ n$ are the dilations $D_{L_n}\phi$ of a fixed function $\phi \in L_1(\mathbb R)$ with $\lim\limits_{n \to \infty} L_n = \infty$ . In this case, the choice of $(D_{L_{n_k}})$ sufficient for the almost everywhere convergence of $(D_{L_{n_k}}\phi\star f)$ for all $f \in L_2(\mathbb R)$ cannot be made to be independent of the function $φ$ .