Geodesic
Upper estimates for best mean-square approximations for some classes of bivariate functions by Fourier-Chebyshev sums
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 268-278

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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: mean-squared approximation, generalized modulus of continuity, Fourier — Tchebychev double series, translated operator. 
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     title = {Upper estimates for best mean-square approximations for some classes of bivariate functions by {Fourier-Chebyshev} sums},
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M. Sh. Shabozov; О. А. Jurakhonov. Upper estimates for best mean-square approximations for some classes of bivariate functions by Fourier-Chebyshev sums. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 268-278. http://geodesic.mathdoc.fr/item/TIMM_2020_26_4_a18/