Sharp estimates for the rate of convergence of double Fourier series in classical orthogonal polynomials

Sharp estimates are obtained for the convergence rate of “triangular” and “hyperbolic” partial sums of Fourier series in orthogonal (Laguerre, Hermite, Jacobi) polynomials in the classes of differentiable functions of two variables characterized by a generalized modulus of continuity. The proofs are based on the generalized shift operator and generalized modulus of continuity for functions from $\mathbb{L}_2$ having generalized partial derivatives in Levi’s sense.