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

Abstract When n) is a sequence of positive functions in L_1(\mathbb R) such that the convolutions \phi_n\star f converge in L1-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 φn 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.


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