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Mathematics. Mechanics. Informatics

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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

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Approximation of Continuous 2π-Periodic Piecewise Smooth Functions by Discrete Fourier Sums

Akniev G. G., Daghestan Scientific Centre of Russian Academy of Sciences

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. 

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