On the behavior of integrable functions at infinity

TL;DR

This paper studies the asymptotic behavior of sequences of scaled integrable functions in multiple dimensions, characterizing conditions under which these functions tend to zero or have summable magnitudes almost everywhere.

Contribution

It provides a detailed description of classes of scaling sequences that determine the pointwise and summability behavior of integrable functions at infinity.

Findings

Characterization of multiplier sequences for convergence to zero.

Conditions for the summability of scaled functions.

Results hold for Lebesgue integrable functions in  dimensions.

Abstract

We investigate the behavior of sequences $(f(c_nx))$ for Lebesgue integrable functions $f:\mathbb R^d\to\mathbb R$. In particular, we give a~description of classes of multipliers $(c_n)$ and $(d_n)$ such that $f(c_nx)\to0$ or $\sum_{n=1}^\infty|f(d_nx)|<\infty$ for $\lambda$ almost every $x\in\mathbb R^d$.