trigonometric polynomials

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

Authors: Akniev G. G.

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.

Approximation Properties of Dicrete Fourier Sums for Some Piecewise Linear Functions

Authors: Akniev G. G.

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.

Orthogonal Basis of Shifts in Space of Trigonometric Polynomials

Authors: Lukashenko T. P.

The orthonormal basis of a system of shifts of one trigonometric polynomial exist in the space of complex trigonometric polynomials with components from m to n and in the space of real trigonometric polynomials with components from 0 to n. Under condition 0 < m < n there is no orthogonal basis of shifts of one trigonometric polynomial in this space real trigonometric polynomials with components from m to n. The system of shifts of two trigonometric polynomials are orthogonal basis in this space.

One counterexample of shape-preserving approximation

Authors: Pleshakov M. G., Tyshkevich S. V.

Let 2s points yi=−π≤y2s<… Example. For each k∈N, k>2, and n∈N there a function f(x):=f(x;s,Y,n,k) exists, such that f∈Δ(1)(Y) and