Geodesic
Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight
Matematičeskie zametki, Tome 110 (2021) no. 3, pp. 434-449.

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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, nonuniform grid, approximation properties.
Mots-clés : Fourier sum, Lebesgue function, orthogonal polynomials 
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     author = {M. S. Sultanakhmedov},
     title = {Approximation of {Functions} by {Discrete} {Fourier} {Sums} in {Polynomials} {Orthogonal} on a {Nonuniform} {Grid} with {Jacobi} {Weight}},
     journal = {Matemati\v{c}eskie zametki},
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     number = {3},
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M. S. Sultanakhmedov. Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight. Matematičeskie zametki, Tome 110 (2021) no. 3, pp. 434-449. http://geodesic.mathdoc.fr/item/MZM_2021_110_3_a8/

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