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 [1].

References

[1]	W. [V.V. Stepanov] Stepanoff, "Sur quelques généralisations des fonctions presque périodiques" C.R. Acad. Sci. Paris , 181 (1925) pp. 90–92

Comments

See also Almost-periodic function.

The different spaces $S_l^p$, each with its metric $D_{S_l^p}$, are topologically equivalent.

References

[a1]	W. [V.V. Stepanov] Stepanoff, "Ueber einige Verallgemeinerungen der fastperiodischen Funktionen" Math. Ann. , 45 (1925) pp. 473–498