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 Mat. Zametki, 2021, Volume 110, Issue 3, Pages 434–449 (Mi mz12833)

Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight

M. S. Sultanakhmedov

Daghestan Scientific Centre of Russian Academy of Sciences, Makhachkala

Abstract: Let there be given a partition of the closed interval $[-1,1]$ by arbitrary nodes $\{\eta_j\}_{j=0}^N$, where $\lambda_N=\max_{0\le j \le N-1} (\eta_{j+1}-\eta_{j})$. For a continuous function $f(t)$ given on an arbitrary grid $\Omega_N=\{t_j \mid \eta_{j} \le t_j \le \eta_{j+1}\}_{j=0}^{N-1}$, the approximation properties of the discrete Fourier sums $\Lambda^{\alpha,\beta}_{n,N}(f,t)$ in polynomials $\widehat P^{\alpha,\beta}_{n, N} (t)$ are investigated in the case of nonnegative integer parameters $\alpha$, $\beta$; these polynomials are orthogonal to $\Omega_N$ with Jacobi weight $\kappa^{\alpha,\beta}(t)=(1-t)^{\alpha}(1+t)^{\beta}$. Given the restriction $n=O(\lambda_N^{-1/3})$ on the order of the Fourier sums, a pointwise estimate of the Lebesgue function $L^{\alpha,\beta}_{n, N}(t)$ is obtained; it depends on $n$ and the position of the point $t \in [-1,1]$:
$$L^{\alpha,\beta}_{n,N}(t)=O[\ln{(n+1)}+ |\widehat P^{\alpha,\beta}_{n,N}(t)|+ |\widehat P^{\alpha,\beta}_{n+1,N}(t)|].$$

Keywords: Jacobi polynomials, Fourier sum, nonuniform grid, Lebesgue function, orthogonal polynomials, approximation properties.

DOI: https://doi.org/10.4213/mzm12833

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English version:
Mathematical Notes, 2021, 110:3, 418–431

Bibliographic databases:

UDC: 517.518.82+517.521
Revised: 04.03.2021

Citation: M. S. Sultanakhmedov, “Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight”, Mat. Zametki, 110:3 (2021), 434–449; Math. Notes, 110:3 (2021), 418–431

Citation in format AMSBIB
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