Harmonic analysis of periodic at infinity functions from Stepanov spaces

I. I. Strukova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 2, pp. 172-182

Résumé

We consider Stepanov spaces of functions defined on $\mathbb{R}$ with their values in a complex Banach space. We introduce the notions of slowly varying and periodic at infinity functions from Stepanov space. The main results of the article are concerned with harmonic analysis of periodic at infinity functions from Stepanov space. For this class of functions we introduce the notion of a generalized Fourier series; the Fourier coefficients in this case may not be constants, they are functions that are slowly varying at infinity. We prove analogs of the classical results on Ćesaro summability. Basic results are derived with the use of isometric representations theory.