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Akniev G. G. Approximation of Continuous 2π-Periodic Piecewise Smooth Functions by Discrete Fourier Sums. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2019, vol. 19, iss. 1, pp. 4-15. DOI: 10.18500/1816-9791-2019-19-1-4-15, EDN: NBHJTJ
Approximation of Continuous 2π-Periodic Piecewise Smooth Functions by Discrete Fourier Sums
Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2πk/N + u (0 6 k 6 N − 1), and denote by Ln,N(f) = Ln,N(f,x) (1 6 n 6 N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk}N−1 k=0 . Select m + 1 points −π = a0 < a1 < ... < am−1 < am = π, where m > 2, and denote Ω = {ai}m i=0. Denote by Cr Ω a class of 2π-periodic continuous functions f, where f is r-times differentiable on each segment ∆i = [ai,ai+1] and f(r) is absolutely continuous on ∆i. In the present article we consider the problem of approximation of functions f ∈ C2 Ω by the polynomials Ln,N(f,x). We show that instead of the estimate |f(x)−Ln,N(f,x)| 6 clnn/n, which follows from the well-known Lebesgue inequality, we found an exact order estimate |f(x)−Ln,N(f,x)| 6 c/n (x ∈ R) which is uniform with respect to n (1 6 n 6 N/2). Moreover, we found a local estimate |f(x)−Ln,N(f,x)| 6 c(ε)/n2 (|x−ai| > ε) which is also uniform with respect to n (1 6 n 6 N/2). The proofs of these estimations are based on comparing of approximating properties of discrete and continuous finite Fourier series.
- Bernshtein S. N. O trigonometricheskom interpolirovanii po sposobu naimen’shih kvadratov [On trigonometric interpolation by the method of least squares]. Dokl. Akad. Nauk USSR, 1934, vol. 4, pp. 1–5 (in Russian).
- Erd¨os P. Some theorems and remarks on interpolation. Acta Sci. Math. (Szeged), 1950, vol. 12, pp. 11–17.
- Kalashnikov M. D. O polinomah nailuchshego (kvadraticheskogo) priblizheniya v zadannoy sisteme tochek [On polynomials of best (quadratic) approximation on a given system of points]. Dokl. Akad. Nauk USSR, 1955, vol. 105, pp. 634–636 (in Russian).
- Krilov V. I. Shodimost algebraicheskogo interpolirovaniya po kornyam mnogochlena Chebisheva dlya absolutno neprerivnih funkciy i funkciy s ogranichennim izmeneniyem [Convergence of algebraic interpolation with respect to the roots of a Chebyshev polynomial for absolutely continuous functions and functions with bounded variation]. Dokl. Akad. Nauk USSR, 1956, vol. 107, pp. 362–365 (in Russian).
- Marcinkiewicz J. Quelques remarques sur l’interpolation. Acta Sci. Math. (Szeged), 1936, vol. 8, pp. 127–130 (in French).
- Marcinkiewicz J. Sur la divergence des polynˆomes d’interpolation. Acta Sci. Math. (Szeged), 1936, vol. 8, pp. 131–135 (in French).
- Natanson I. P. On the Convergence of Trigonometrical Interpolation at Equi-Distant Knots. Annals of Mathematics, Second Ser., 1944, vol. 45, no. 3, pp. 457–471. DOI: http://doi.org/10.2307/1969188
- Nikolski S. M. Sur certaines methodes d’approximation au moyen de sommes trigonome?triques. Izv. Akad. Nauk SSSR, Ser. Mat., 1940, vol. 4, iss. 4, pp. 509–520 (in Russian).
- Turetskiy A. H. Teorija interpolirovanija v zadachah [Interpolation theory in exercises]. Minsk, Vissheyshaya Shkola Publ., 1968. 320 p. (in Russian).
- Zygmund A. Trigonometric Series. Vol. 1. Cambridge, Cambridge Univ. Press, 1959. 747 p.
- Akniyev G. G. Discrete least squares approximation of piecewise-linear functions by trigonometric polynomials. Issues Anal., 2017, vol. 6 (24), iss. 2, pp. 3–24. DOI: http://doi.org/10.15393/j3.art.2017.4070
- Sharapudinov I. I. On the best approximation and polynomials of the least quadratic deviation. Anal. Math., 1983, vol. 9, iss. 3, pp. 223–234.
- Sharapudinov I. I. Overlapping transformations for approximation of continuous functions by means of repeated mean Valle Poussin. Daghestan Electronic Mathematical Reports, 2017, iss. 8, pp. 70–92.
- Courant R. Differential and Integral Calculus. Vol. 1. New Jersey, Wiley-Interscience, 1988. 704 p.
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