Optimal recovery of periodic functions from Fourier coefficients given with an error
The paper is concerned with optimal recovery of linear functionals on Hilbert spaces based on approximate Fourier coefficients with a prescribed error in the uniform norm. The general framework is the following: Let \(X\) be a Hilbert space with a complete orthonormal system \((e_j,j\in\mathbb{N})\) and let \(x_j=(x,e_j)\), \(j\in\mathbb{N}\), denote the Fourier coefficients of \(x\). Let also \(B_X\) be the closed unit ball of \(X\). For \(f\in X\) an optimal recovery of the functional \((x,f)\), \(x\in B_X\), from approximate Fourier coefficients \(\widetilde x_j\) verifying \(|x_j-\widetilde x_j|\leq \delta_j\), \(j=1,\dots, n\), is given by \(\sum^n_{j=1} (1-\lambda\delta_j|f_j|^{-1})_+\overline f_j\widetilde x_j\), where \(\lambda\in(0,|f|)\) is the solution of the equation \(|f|^2- \sum^n_{j=1} (|f_j|^2-\lambda^2\delta^2_j)_+-\lambda^2=0\). (Here, as usual, \(a_+=2^{-1}(|a|+a))\). The optimal value of the method is \(e(f,\delta)=\lambda+ \sum^n_{j=1} \delta_j(|f_j|-\lambda\delta_j)_+\) (Theorem 1). The obtained results are applied to optimal recovery of \(2\pi\)-periodic functions and of their derivatives in Hardy-Sobolev and Bergman-Sobolev spaces. A similar problem, but for exact values of the Fourier coefficients, was solved by \textit{B. D. Boyanov} [Serdica 2, 300-304 (1976; Zbl 0442.42005)].
Hardy-Sobolev and Bergman-Sobolev spaces
scientific article; zbMATH DE number 884756