On the behavior of integrable functions at infinity

Abstract

We investigate the behavior of sequences $(f(c_{n}x))$ 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 $f(c_{n}x)\to 0$ or ${\sum }_{n=1}^{\infty }|f(d_{n}x)|<\infty$ for $\lambda$ almost every $x\in {\mathbb{R}}^d$.