Cite this article as:
Akniev G. G. Approximation of Continuous 2 p-Periodic Piecewise Smooth Functions by Discrete Fourier Sums . Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2019, vol. 19, iss. 1, pp. 4-15. DOI: https://doi.org/10.18500/1816-9791-2019-19-1-4-15
Approximation of Continuous 2 p-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.
1. 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).
2. Erd¨os P. Some theorems and remarks on interpolation. Acta Sci. Math. (Szeged), 1950, vol. 12, pp. 11–17.
3. 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).
4. 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).
5. Marcinkiewicz J. Quelques remarques sur l’interpolation. Acta Sci. Math. (Szeged), 1936, vol. 8, pp. 127–130 (in French).
6. Marcinkiewicz J. Sur la divergence des polynˆomes d’interpolation. Acta Sci. Math. (Szeged), 1936, vol. 8, pp. 131–135 (in French).
7. 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
8. 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).
9. Turetskiy A. H. Teorija interpolirovanija v zadachah [Interpolation theory in exercises]. Minsk, Vissheyshaya Shkola Publ., 1968. 320 p. (in Russian).
10. Zygmund A. Trigonometric Series. Vol. 1. Cambridge, Cambridge Univ. Press, 1959. 747 p.
11. 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
12. Sharapudinov I. I. On the best approximation and polynomials of the least quadratic deviation. Anal. Math., 1983, vol. 9, iss. 3, pp. 223–234.
13. 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.
14. Courant R. Differential and Integral Calculus. Vol. 1. New Jersey, Wiley-Interscience, 1988. 704 p.
