Approximation Properties of Discrete Fourier Sums in Polynomials Orthogonal on Non-Uniform Grids

Nurmagomedov, A. A.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 2.

Abstract:
Given two positive integers \(\alpha\) and \(\beta\), for arbitrary continuous function \(f(x)\) on the segment \([-1, 1]\) we construct disrete Fourier sums \(S_{n,N}^{\alpha,\beta}(f,x)\) on system polynomials \(\big\{\hat{p}_{k,N}^{\alpha,\beta}(x)\big\}_{k=0}^{N-1}\) forming an orthonormals system on any finite non-uniform set \(\Omega_N=\{x_j\}_{j=0}^{N-1}\) of \(N\) points from segment \([-1, 1]\) with Jacobi type weight. The approximation properties of the corresponding partial sums \(S_{n,N}^{\alpha,\beta}(f,x)\) of order \(n\leq{N-1}\) in the space of continuous functions \(C[-1, 1]\) are investigated. Namely, for a Lebesgue function in \(L_{n,N}^{\alpha,\beta}(x)\), a two-sided pointwise estimate of discrete Fourier sums with \(n=O\Big(\delta_N^{-\frac{1}{(\lambda+3)}}\Big)\), \(\lambda=\max\{\alpha, \beta\}\), \(\delta_N=\max_{0\leq{j}\leq{N-1}}\Delta{t_j}\) is obtained. The problem of convergence of \(S_{n,N}^{\alpha,\beta}(f,x)\) to \(f(x)\) is also investigated. In particular, an estimate is obtained of the deviation of the partial sum \(S_{n,N}^{\alpha,\beta}(f,x)\) from \(f(x)\) for \(n=O\Big(\delta_N^{-\frac{1}{(\lambda+3)}}\Big)\), depending on \(n\) and the position of a point \(x\) in \([-1, 1].\)

Keywords: polynomial, orthogonal system, net, weight, asymptotic formula, Fourier sum, Lebesgue function.

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