On approximation of continuous functions by trigonometric polynomials

Abstract

We generalize the classical Jackson-Bernstein constructive description of Hölder classes of periodic functions on the interval [−π, π]. We approximate by trigonometric polynomials continuous functions defined on a compact set E ⊂ [−π, π]. This set may consist of an infinite number of intervals. Published by Elsevier Inc.

The Trigonometric Polynomial Like Bernstein Polynomial

“…Quasi-interpolant on trigonometric splines has been discussed in [ 14 ]. In [ 15 ], authors approximate continuous functions defined on a compact set E ∈ [− π , π ] by trigonometric polynomials. Some problems of geometric modeling are solved better by trigonometric splines.…”

Weighted fuzzy transform and its application for approximation of discrete functions by continuous functions

“…During the past decade many researchers have been done many investigation that how functions can be approximated with simpler functions [1,7]. Fuzzy transform is one of these methods.…”

“…Let f (x) = 2 cos(4x) on the interval[1,3]. Consider two weight functions defined by ϕ 1 (x) = e −x and ϕ 2 (x) = x+1 4 .…”

“…Consider two weight functions defined by ϕ 1 (x) = e −x and ϕ 2 (x) = x+1 4 . We want to approximate the function f on the interval[1,3] by these two different weight functions and then compare the approximations with each other. Let the weighted fuzzy partition be B k (x) = ϕ(x)A k (x) in each case, where A k (x) is the triangular membership function, i.e.…”