Recovery of continuous functions of two variables from their Fourier coefficients known with error

Keywords:
Fourier series, method of regularization, ΛΛ-method of summationAbstract
In this paper, we continue to study the classical problem of optimal recovery for the classes of continuous functions. The investigated classes Wψ2,pWψ2,p, 1≤p<∞1≤p<∞, consist of functions that are given in terms of generalized smoothness ψψ. Namely, we consider the two-dimensional case which complements the recent results from [Res. Math. 2020, 28 (2), 24-34] for the classes WψpWψp of univariate functions.
As to available information, we are given the noisy Fourier coefficients yδi,j=yi,j+δξi,jyδi,j=yi,j+δξi,j, δ∈(0,1)δ∈(0,1), i,j=1,2,…i,j=1,2,…, of functions with respect to certain orthonormal system {φi,j}∞i,j=1{φi,j}∞i,j=1, where the noise level is small in the sense of the norm of the space lplp, 1≤p<∞1≤p<∞, of double sequences ξ=(ξi,j)∞i,j=1ξ=(ξi,j)∞i,j=1 of real numbers. As a recovery method, we use the so-called ΛΛ-method of summation given by certain two-dimensional triangular numerical matrix Λ={λni,j}ni,j=1Λ={λni,j}ni,j=1, where nn is a natural number associated with the sequence ψψ that define smoothness of the investigated functions. The recovery error is estimated in the norm of the space C([0,1]2)C([0,1]2) of continuous on [0,1]2[0,1]2 functions.
We showed, that for 1≤p<∞1≤p<∞, under the respective assumptions on the smoothness parameter ψψ and the elements of the matrix ΛΛ, it holds Δ(Wψ2,p,Λ,lp)=supy∈Wψ2,psup‖ξ‖lp≤1‖y−n∑i=1n∑j=1λni,j(yi,j+δξi,j)φi,j‖C([0,1]2)≪nβ+1−1/pψ(n).Δ(Wψ2,p,Λ,lp)=supy∈Wψ2,psup∥ξ∥lp≤1∥∥y−n∑i=1n∑j=1λni,j(yi,j+δξi,j)φi,j∥∥C([0,1]2)≪nβ+1−1/pψ(n).