Recovery of continuous functions of two variables from their Fourier coefficients known with error
In this paper, we continue to study the classical problem of optimal recovery for the classes of continuous functions. The investigated classes W^ψ_2,p, 1 ≤ p < ∞, consist of functions that are given in terms of generalized smoothness ψ. Namely, we consider the two-dimensional case which complements the recent results from [Res. Math. 2020, 28 (2), 24-34] for the classes W^ψ_p of univariate functions. As to available information, we are given the noisy Fourier coefficients y^δ_i,j = y_i,j + δ ξ_i,j, δ (0,1), i,j = 1,2, …, of functions with respect to certain orthonormal system { φ_i,j }_i,j=1^∞, where the noise level is small in the sense of the norm of the space l_p, 1 ≤ p < ∞, of double sequences ξ=( ξ_i,j )_i,j=1^∞ of real numbers. As a recovery method, we use the so-called Λ-method of summation given by certain two-dimensional triangular numerical matrix Λ = { λ_i,j^n }_i,j=1^n, where n is a natural number associated with the sequence ψ that define smoothness of the investigated functions. The recovery error is estimated in the norm of the space C ([0,1]^2) of continuous on [0,1]^2 functions. We showed, that for 1≤ p < ∞, under the respective assumptions on the smoothness parameter ψ and the elements of the matrix Λ, it holds \[ Δ( W^ψ_2,p, Λ, l_p)= _ y W^ψ_2,p _\| ξ \|_l_p ≤ 1 \| y - _i=1^n _j=1^n λ_i,j^n ( y_i,j + δ ξ_i,j) φ_i,j \|_C ([0,1]^2) n^β + 1 - 1/p ψ(n).\] , number=3, journal=Carpathian Mathematical Publications, author=Pozharska, K.V. and Pozharskyi, A.A., year=2021, month=Dec., pages=676–686