Bounds for Fourier widths of classes of periodic functions with a mixed modulus of smoothness

Sh. A. Balgimbaeva   ; T. I. Smirnov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 4, pp. 78-94

Résumé

Order-exact bounds are obtained for Fourier widths of the Nikol'skii-Besov classes $\mathrm{SB}_{p\theta}^{\Omega,l} (\mathbb{T}^d)$ and Triebel-Lizorkin classes $\mathrm{SF}_{p\theta}^{\Omega,l} (\mathbb{T}^d)$ of functions with a given majorant $\Omega$ for the mixed modulus of smoothness of order $l$ in the space $L_q(\mathbb{T}^d)$ for all relations between the parameters $p$, $q$, and $\theta$ under some conditions on $\Omega$. The upper bounds follow from order-exact bounds for approximations of the classes $\mathrm{SB}_{p\theta}^{\Omega,l} (\mathbb{T}^d)$ and $\mathrm{SF}_{p\theta}^{\Omega,l} (\mathbb{T}^d)$ by special partial sums of Fourier series in the multiple system $\Psi_d$ of periodized Meyer wavelets.

Détail

Keywords: fourier width, mixed modulus of smoothness, function spaces, wavelet system.

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