On the behavior of integrable functions at infinity

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 $x\in\mathbb R^d$.