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 Vladikavkaz. Mat. Zh., 2020, Volume 22, Number 2, Pages 34–47 (Mi vmj722)

Approximation properties of discrete Fourier sums in polynomials orthogonal on non-uniform grids

A. A. Nurmagomedov

Dagestan State Agrarian University, 180 M. Gadzhiev St., Makhachkala 367032, Russian

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 $\{\hat{p}_{k,N}^{\alpha,\beta}(x)\}_{k=0}^{N-1}$ 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 $n=O(\delta_N^{-\frac{1}{(\lambda+3)}})$, $\lambda=\max\{\alpha, \beta\}$, $\delta_N=\max_{0\leq{j}\leq{N-1}}\Delta{t_j}$ 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 $n=O(\delta_N^{-\frac{1}{(\lambda+3)}})$, depending on $n$ and the position of a point $x$ in $[-1, 1].$

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

DOI: https://doi.org/10.46698/k4355-6603-4655-y

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UDC: 517.98
MSC: 42C10

Citation: A. A. Nurmagomedov, “Approximation properties of discrete Fourier sums in polynomials orthogonal on non-uniform grids”, Vladikavkaz. Mat. Zh., 22:2 (2020), 34–47

Citation in format AMSBIB
\Bibitem{Nur20} \by A.~A.~Nurmagomedov \paper Approximation properties of discrete Fourier sums in polynomials orthogonal on non-uniform grids \jour Vladikavkaz. Mat. Zh. \yr 2020 \vol 22 \issue 2 \pages 34--47 \mathnet{http://mi.mathnet.ru/vmj722} \crossref{https://doi.org/10.46698/k4355-6603-4655-y}