Upper estimates for best mean-square approximations for some classes of bivariate functions by Fourier-Chebyshev sums
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.