M.Sh. Shabozov, O.A. Dzhurakhonov. Approximation in the mean of some classes of bivariate functions by Fourier–Chebyshev sums ... P. 268-278

In space $L_{2,\rho}$ of bivariate functions summable with square on set $Q=[-1,1]^2$ with weight $\rho(x,y)={1}/{\sqrt{(1-x^{2})(1-y^{2})}}$ the sharp inequalities of Jackson--Stechkin type in which the best polynomial approximation estimated above by Peetre $\mathcal{K}$-functional were obtained. We also find the exact values of various widths of classes of functions defined by generalized modulus of continuity and $\mathcal{K}$-functionals. Also the exact upper bounds for modules of coefficients of Fourier--Tchebychev on considered classes of functions were calculated.

Keywords: approximation, generalized modulus of continuity, Fourier–Chebyshev double series, generalized translation operator

Received August 08, 2020

Revised November 16, 2020

Accepted November 23, 2020

Mirgand Shabozovich Shabozov, Dr. Sci. (Phys.-Math.), Prof., member of Academy of NAN Tajikistan, Tajik National University, Dushanbe, 734025 Republic of Tajikistan, e-mail: shabozov@mail.ru

Olimjon Akmalovich Jurakhonov, Tajik National University, Dushanbe, Associate Professor of the Department of Functional Analysis and Differential Equations, 734025 Republic of Tajikistan, e-mail: olim1974@mail.ru

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Cite this article as: M.Sh. Shabozov, O.A. Dzhurakhonov. Approximation in the mean of some classes of bivariate functions by Fourier–Chebyshev sums, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 4, pp. 268–278.