Conditions of absolute cesaro summability of multiple trigonometric Fourier series

S. Bitimkhan

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 2, pp. 42-47

Résumé

A necessary and sufficient condition of absolute $|C;\overline{\beta}|_\lambda$-summability almost everywhere on ${\mathbb T}^s$ is obtained for multiple trigonometric Fourier series of functions $f\in L_{\overline{q}}({\mathbb T}^s)$ from generalized Besov classes $B_{\overline q,s,\theta}^{\omega_r}$, where ${\mathbb T}^s=[0,2\pi)^s$, $\overline{\beta}=(\beta_1,\beta_2,\ldots,\beta_s)$, $\overline{q}=(q_1,q_2,\ldots, q_s)$, $1$, $\overline{1,s}$, $1\le \lambda\le q_s\le \ldots\le q_1$, $\lambda\theta\infty$, $0\le \beta_j1/q'_j=1-1/q_j$, $\overline{1,s}$, $r\in \mathbb{N}$, $r>\sum_{j=1}^s(1/q_j-\beta_j)$, and $\omega_r$ is a function of the type of modulus of smoothness of order $r$.

Détail

Keywords: multiple trigonometric Fourier series, absolute summability, modulus of smoothness, generalized Besov class.