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

Authors

  • K.V. Pozharska Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska str., 01601, Kyiv, Ukraine https://orcid.org/0000-0001-7599-8117
  • A.A. Pozharskyi Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereschenkivska str., 01601, Kyiv, Ukraine
https://doi.org/10.15330/cmp.13.3.676-686

Keywords:

Fourier series, method of regularization, ΛΛ-method of summation
Published online: 2021-12-10

Abstract

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, 1p<1p<, 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, 1p<1p<, 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 1p<1p<, under the respective assumptions on the smoothness parameter ψψ and the elements of the matrix ΛΛ, it holds Δ(Wψ2,p,Λ,lp)=supyWψ2,psupξlp1yni=1nj=1λni,j(yi,j+δξi,j)φi,jC([0,1]2)nβ+11/pψ(n).Δ(Wψ2,p,Λ,lp)=supyWψ2,psupξlp1yni=1nj=1λni,j(yi,j+δξi,j)φi,jC([0,1]2)nβ+11/pψ(n).

How to Cite
(1)
Pozharska, K.; Pozharskyi, A. Recovery of Continuous Functions of Two Variables from Their Fourier Coefficients Known With Error. Carpathian Math. Publ. 2021, 13, 676-686.