Abstract: Let \(T=\{t_0, t_1, \ldots, t_N\}\) and \(T_N=\{x_0, x_1, \ldots, x_{N-1}\},\) where \(x_j=(t_j+t_{j+1})/2\), \(j=0, 1, \ldots, N-1,\) are any system of different points from \([-1, 1].\) For arbitrary continuous function \(f(x)\) on the segment \([-1, 1]\) we construct Valle-Poussin type averages \(V_{n,m,N}(f,x)\) for discrete Fourier sums \(S_{n,N}(f,x)\) on system of polynomials \(\{\hat{p}_{k,N}(x)\}_{k=0}^{N-1}\) forming an orthonormals system on any finite non-uniform grids \(T_N=\{x_j\}_{j=0}^{N-1}\) with weight \(\Delta{t_j}=t_{j+1}-t_j.\) Approximation properties of the constructed averages \(V_{n,m,N}(f,x)\) of order \(n+m\leq{N-1}\) in the space of continuous functions \(C[-1, 1]\) are investigated. Namely, it is proved that the Vallee-Poussin averages \(V_{m,n,N}(f,x)\) for \(\frac{n}m\asymp1, n\leq\lambda\delta_N^{-\frac14} (\lambda>0), \delta_N=\max_{0\leq{j}\leq{N-1}}\Delta{t_j},\) are uniformly bounded as a family of linear operators acting in the space \(C[-1, 1].\) In addition, as a consequence of the obtained result the order of approximation of the continuous function \(f(x)\) by the Vallee-Poussin \(V_{n,m,N}(f,x)\) averages in space \(C[-1, 1]\) is established.
For citation: Nurmagomedov, A. A. and Shikhshinatova, M. M. Approximation Properties of Valle-Poussin Averages for Discrete Fourier Sums Bypolynomials Orthogonal on Arbitrary Nets, Vladikavkaz Math. J., 2025, vol. 27, no. 2, pp. 93-111 (in Russian). DOI 10.46698/q4030-9541-4914-r
1. Daugavet, I. K. and Rafalson, S. Z. About of Some Inequalities for
Algebraic Polynomial, Vestnik Leningradskogo universiteta} [Bulletin of Leningrad University], 1974, vol. 19, pp. 18-24 (in Russian).
2. Konyagin, S. V. Markov's Inequality for Polynomials in the
Metric of \(L\), Proceedings of the Steklov Institute of Mathematics, 1981, vol. 145, pp. 129-138.
3. Simonov, I. E. Sharp Markov Brothers Type Inequality in the
Spaces \(L_p, L_1\) on the a Closed Interval, Trudy Instituta Matematiki i Mekhaniki UrO RAN [Proceedings of the Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences], 2011, vol. 17, no. 3 pp. 282-290 (in Russian).
4. Nurmagomedov, A. A. and Rasulov, N. K. Two-Sided Estimates of
Fourier Sums Lebesgue Functions with Respect to Polynomials
Orthogonal on Nonuniform Grids, Vestnik St. Petersburg
University, Mathematics, 2018, vol. 51, no. 3, pp. 249-259. DOI:
10.3103/S1063454118030068.
5. Nurmagomedov, A. A. The Approximation of Functions by Partial Sums of the Fourier Series in Polynomials Orthogonal on Arbitrary Grid, Russian Mathematics (Izvestiya VUZ. Matematika), 2020, vol. 64, no. 4, pp. 54-63. DOI: 10.3103/S1066369X20040064.
6. Nurmagomedov, A. A. Approximation Properties of Discrete Fourier Sums in Polynomials Orthogonal on Non-Uniform Grids, Vladikavkaz Mathematical Journal, 2020, vol. 22, no. 2, pp. 34-47 (in Russian). DOI: 10.46698/k4355-6603-4655-у.
7. Sharapudinov, I. I. and Vagabov, I. A. Convergence of the
Vallee-Poussin Means for Fourier-Jacobi Sums, Mathematical Notes,
1996, vol. 60, iss. 4, pp. 425-437. DOI: 10.1007/BF02305425.
8. Sharapudinov, I. I. Boundednes in \(C[- 1, 1]\) of the de la Vallee-Poussin Means
for the Discrete Fourier-Jacobi Sums, Sbornik: Mathematics,
1996, vol. 187, no. 1, pp. 141-158. DOI: 10.1070/SM1996v187n01ABEH000105.
9. Sharapudinov, I. I. Approximation Properties of the Vallee-Poussin Means of Partial Sums of a Mixed Series of Legendre Polynomials, Mathematical Notes, 2008, vol. 84, no. 3,
pp. 417-434. DOI: 10.1134/ S0001434608090125.
10. Korkmasov, F. M. Approximation Properties of the de la
Vallee-Poussin Means for Discrete Fourier-Jacobi Sums, Siberian Mathematical Journal,
2004, vol. 45, no. 2, pp. 273-293. DOI: 10.1023/B:SIMJ.0000021284.60159.bd.
11. Nurmagomedov, A. A. Asymptotic Properties of Polynomials which
are Orthogonal on Arbitrary Grids, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2008, vol. 8, no. 1, pp. 25-31 (in Russian). DOI: 10.18500/1816-9791-2008-8-1-25-31.
12. Szego, G. Orthogonal Polynomials, American Mathematical Society, 1939, 432 p.
13. Suetin, P. K. Klassicheskie ortogonal'nye mnogochleny [Classical Orthogonal Polynomials], Moscow, Nauka, 1979 (in Russian).
14. Sharapudinov, I. I. Mnogochleny, ortogonal'nye na setkah. Teoriya i prilozheniya [Mixed Series of Orthogonal
Polynomials. Theory and Applications], Makhachkala, Dagestan State Pedagogical University, 1997, 255 p. (in Russian).
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