Approximation of functions by a hyperbolic angle in the \(L_2\)-metric (Q1569791)

In the \(L^2\) space of the product \(X:=X_1\times X_2\) of two measurable spaces the spaces \(H_{kl}\) of harmonics \(Y_{kl}\) of order \(k,l\) are defined in terms of a complete orthonormal system. Generalized operators of differentiation and translation are introduced by means of Fourier expansion of a function \(f\) with respect to the system \(Y_{kl}\). With these operators logarithmic classes similar to the Nikol'skii classes are defined. For functions in these classes the degree of approximation in the \(L^2\)-norm by harmonics lying in an angle \(\{(k,l): k < n_1\) or \(l < n_2\}\) or in an hyperbolic angle \(\{(k,l): k^{\rho_1}l^{\rho_2}<N\}\) with parameters \(\rho_i >0\) is estimated, and direct and inverse theorems are stated.