# Stepanov almost-periodic functions

A class $S_l^p$ of functions that are measurable and summable together with their $p$-th power $(p\geq1)$ on every finite interval $[x,x+l]$ and that can be approximated in the metric of the Stepanov space (see below) by finite sums
$$\sum_{n=1}^Na_ne^{i\lambda_nx},$$
where $a_n$ are complex coefficients and $\lambda_n$ are real numbers. The distance in the Stepanov space is defined by the formula
$$D_{S_l^p}[f(x),g(x)]=\sup_{-\infty<x<\infty}\left[\frac1l\int\limits_x^{x+l}|f(x)-g(x)|^pdx\right]^{1/p}.$$
Functions of the class $S_l^p$ can also be defined using the concept of an almost-period.
Functions of the class $S^p=S_1^p$ possess a number of properties also possessed by Bohr almost-periodic functions. For example, functions of the class $S^p$ are bounded and uniformly continuous (in the metric $D_{S_l^p}$), the limit $f$ of a convergent sequence of Stepanov almost-periodic functions $\{f_n\}$ (in the metric of $S^p$) belongs to $S^p$. If a function in $S^p$ is uniformly continuous (in the ordinary sense) on the whole real axis, then it is a Bohr almost-periodic function. Introduced by V.V. Stepanov .