Pointwise Convergence of Sequences of Singular Measures

TL;DR

This paper studies the pointwise convergence of convolution operators generated by probability measures on the real line, identifying conditions under which these operators approximate functions almost everywhere.

Contribution

It establishes conditions for almost everywhere convergence of convolution sequences to functions within specific classes, focusing on contractions of a single measure as approximate identities.

Findings

Convergence occurs under certain regularity conditions on measures.

Results apply to sequences of contractions of a fixed probability measure.

Provides criteria for pointwise convergence in the context of approximate identities.

Abstract

We investigate the almost everywhere convergence of sequences of convolution operators given by probability measures $\mu_n$ on $\mathbb R$. If this sequence of operators constitutes an approximate identity on a particular class of functions $\mathcal F$, under what additional conditions do we have $\mu_n \ast f \to f$ a.e. for all $f \in \mathcal F$? We focus on the particular case of a sequence of contractions $C_{t_n}\mu$ of a single probability measure $\mu$, with $t_n \to 0$, so that that the sequence of operators is an approximate identity.