Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials

Two-variable functions f ( x , y ) from the class L 2 = L 2 (( a , b ) × ( c , d ); p ( x ) q ( y )) with the weight p ( x ) q ( y ) and the norm are approximated by an orthonormal system of orthogonal P n ( x ) Q n ( y ), n , m = 0, 1, ..., with weights p ( x ) and q ( y ). Let denote the best approximation of f ∈ L 2 by algebraic polynomials of the form . Consider a double Fourier series of f ∈ L 2 in the polynomials P n ( x ) Q m ( y ), n , m = 0, 1, ..., and its “hyperbolic” partial sums A generalized shift operator Fh and a k th-order generalized modulus of continuity Ω k ( A , h ) of a function f ∈ L 2 are used to prove the following sharp estimate for the convergence rate of the approximation: . Moreover, for every fixed N = 4, 9, 16, ..., the constant on the right-hand side of this inequality is cannot be reduced.