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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 $t \in [-1,1]$: $$ L^{\alpha,\beta}_{n,N}(t)=O\bigl[\ln{(n+1)}+ |\widehat P^{\alpha,\beta}_{n,N}(t)|+ |\widehat P^{\alpha,\beta}_{n+1,N}(t)|\bigr]. $$
Keywords:
Jacobi polynomials, Fourier sum, nonuniform grid, Lebesgue function, orthogonal polynomials, approximation properties.
Received: 07.07.2020 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
Linking options:
https://www.mathnet.ru/eng/mzm12833https://doi.org/10.4213/mzm12833 https://www.mathnet.ru/eng/mzm/v110/i3/p434
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Abstract page: | 221 | Full-text PDF : | 28 | References: | 34 | First page: | 2 |
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