A rate $L_p$-version for the Riesz criterion of the absolute convergence of trigonometric Fourier series

N. A. Il'yasov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 4, pp. 193-202

Résumé

Rate versions of the Riesz criterion ($A=L_2(\mathbb T)*L_2(\mathbb T)$) are established in terms of best approximations ($A[\lambda]=E_2t[\lambda^{1/2}]*E_2[\lambda^{1/2}]$) and moduli of smoothness ($A[\omega]=H_2^l[\omega^{1/2}]*H_2^l[\omega^{1/2}]$) of the functions that compose the convolution, and conditions are found for $\lambda$ (necessary and sufficient in the case $1\le p2$ and sufficient in the case $2$) under which the equality$A[\lambda]=E_p[\lambda^{1/2}]*E_p[\lambda^{1/2}]$ is valid, where $\lambda\in M_0$, $\omega\in\Omega_l$, $l\in\mathbb N$.

Détail

Keywords: trigonometric Fourier series, absolute convergence, convolution of two functions, best approximation, modulus of smoothness, rate version of the Riesz criterion.