Approximation of bivariate functions by Fourier–Tchebychev “circular” sums in $L_{2,\rho}$

In this paper the sharp upper bounds of approximation of functions of two variables with generalized Fourier–Chebyshev polynomials for the class of functions $L_{2,\rho}^{(r)} (D)$, $r\in N,$ are calculated in $L_{2,\rho}:=L_{2,\rho}(Q)$, where $\rho:=\rho(x,y)=1/\sqrt{(1-x^{2})(1-y^{2})}$, $Q:=\{(x,y):-1\leq x,y\leq1\}$, and $D$ is a second order Chebyshev–Hermite operator. The sharp estimates for the best polynomial approximation are obtained by means of weighted average of module of continuity of $m$-th order with $D^r f$ $(r\in Z_+)$ in $L_{2,\rho}$. The sharp estimates for the best approximation of double Fourier series in Fourier–Chebyshev orthogonal system in the classes of functions of several variables which are characterized by generalized module of continuity are given. We first form some classes of functions and then the corresponding methods of approximations, “circular” by partial sum of Fourier–Chebyshev double series, since, unlike the one-dimensional case, there is no natural way of expressing the partial sums of double series. The shift operator plays a crucial role in the problems related to expansion of functions in Fourier series in trigonometric system and estimating their best approximation properties. Based on some previous known research we construct the shift operator, which enables one to determine some classes of functions which characterized by module of continuity. And for these classes of functions the upper bound for the best mean squared approximation by “circular” partial sum of Fourier–Chebyshev double series is calculated.