# Convergence of Sequences of Convolution Operators

### From NZJM

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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 (φ_{n}) is a sequence of positive
functions in
such that the convolutions converge in *L*_{1}-norm to
*f* for all , then these convolutions may or may
not converge almost everywhere too. There is always a subsequence
such that at least converges almost
everywhere to *f* for all .
A special case of this is when the functions φ_{n} are the
dilations of a fixed function
with . In this case, the
choice of sufficient for the almost everywhere
convergence of for all cannot
be made to be independent of the function φ.

**Keywords**

**Classification** (MSC2000)

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