A class of functions of a real variable

Yu. I. Alimov

Matematičeskie zametki, Tome 8 (1970) no. 2, pp. 149-158.

Résumé

An investigation of measurable almost-everywhere finite functions $\xi(t)$, $-\infty$, for which $$ \varphi_T^\xi(\tau_{(n)},\lambda_{(n)})=\frac1{2T}\int_{-T}^T\exp{i}\sum_{k=1}^n\lambda_k\xi(t-\tau_k)dt $$ tends to an asymptotic characteristic function $\varphi_\infty^\xi(\tau_{(n)},\lambda_{(n)})$ when $T\to\infty$. Here $n$ is any positive integer and $\tau_{(n)}=(\tau_1,\tau_2,\dots,\tau_n)$ is arbitrary. It is proved that the class of such functions $\xi(t)$ is larger than the class of Besicovich almost-periodic functions.