Distributional Point Values and Delta Sequences

TL;DR

This paper investigates how distributional point values can be characterized using delta sequences, providing new insights into when distributions are regular functions and how limits can be understood through sequence convergence.

Contribution

It offers a novel connection between delta sequence-based evaluations and classical distributional point values, including criteria for regularity and limit characterization.

Findings

Distributional point values can be described via delta sequences.

A criterion for a distribution to be a bounded function.

Limits in continuous variables can be characterized by sequence limits.

Abstract

Recently Sasane defined a notion of evaluating a distribution at a point using delta sequences. In this paper, we explore the relationship between generalizations of his definition and the standard definition of distributional point values. This allows us to obtain a description of distributional point values via delta sequences and a characterization of when a distribution is actually a regular distribution given by bounded function. We also give a characterization of limits in a continuous variable by the existence of the limits of certain sequences.