Cite this article as:
Akniev G. G. Approximation Properties of Dicrete Fourier Sums for Some Piecewise Linear Functions. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2018, vol. 18, iss. 1, pp. 4-16. DOI: https://doi.org/10.18500/1816-9791-2018-18-1-4-16
Approximation Properties of Dicrete Fourier Sums for Some Piecewise Linear Functions
Let N be a natural number greater than 1. We select N uniformly distributed points t_k = 2πk/N (0 < k < N − 1) on [0,2\pi]. Denote by L_ n,N (f) = L _n,N (f,x)1 < n < ⌊N/2⌋ the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system tk{k=0}^{N-1}. In other words, the greatest lower bound of the sums on the set of trigonometric polynomials Tn of order n is attained by L_n,N (f). In the present article the problem of function approximation by the polynomials L_n,N (f,x) is considered. Using some example functions we show that the polynomials Ln,N (f,x) uniformly approximate a piecewise-linear continuous function with a convergence rate O(1/n) with respect to the variables x ∈ R and 1 < n < N/2.These polynomials also uniformly approximate the same function with a rate O(1/n^2) outside of some neighborhood of function’s „crease“points. Also we show that the polynomials Ln,N (f,x) uniformly approximate a piecewise-linear discontinuous function with a rate O(1/n) with respect to the variables x and 1< n < N/2 outside some neighborhood of discontinuity points. Special attention is paid to approximation of 2π-periodic functions f1 and f2 by the polynomials L n,N (f,x), where f1 (x) = |x| and f2 (x) = sign x for x ∈ [−π,π]. For the first function f1 we show that instead of the estimate |f1 (x) − L n,N (f1 ,x)| < clnn/n which follows from the well-known Lebesgue inequality for the polynomials L n,N (f,x) we found an exact order estimate |f1 (x) − L n,N (f1 ,x)| < c/n (x ∈ R) which is uniform relative to 1 < n < N/2. Moreover, we found a local estimate |f1 (x) − L n,N (f1 ,x)| < c(ε)/n 2 (|x − πk| > ε) which is also uniform relative to 1 < n < N/2.For the second function f2 we found only a local estimate |f 2 (x) − L n,N (f2 ,x)| < c(ε)/n (|x − πk| > ε) which is uniform relative to 1 < n < 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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