function approximation

The Principle of Localization at the Class of Functions Integrable in the Riemann for the Processes of Lagrange –Sturm – Liouville

Authors: Trynin A. Y., Kireeva E. D.

Let us say that the principle of localization holds at the class of functions F at point x0 ∈ [0, π] for the Lagrange –Sturm – Liouville interpolation process L SL n (f, x) if limn→∞ L SL n (f, x0) − L SL n (g, x0) = 0 follows from the fact that the condition f(x) = g(x) is met for any two functions f and g belonging to F in some neighborhood Oδ(x0), δ > 0.

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.

Special Wavelets Based on Chebyshev Polynomials of the Second Kind and their Approximative Properties

Authors: Sultanakhmedov M. S.

The system of wavelets and scalar functions based on Chebyshev polynomials of the second kind and their zeros is considered. With the help of them we construct a complete orthonormal system of functions.