The Trigonometric Polynomial Like Bernstein Polynomial

Abstract

A symmetric basis of trigonometric polynomial space is presented. Based on the basis, symmetric trigonometric polynomial approximants like Bernstein polynomials are constructed. Two kinds of nodes are given to show that the trigonometric polynomial sequence is uniformly convergent. The convergence of the derivative of the trigonometric polynomials is shown. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed. Some interesting properties of the trigonometric polynomials are given.

Piecewise trigonometric Hermite interpolation

“…It is not convenient for the representation of a trigonometric Hermite interpolant by using these basis functions. Now we consider the symmetric, nonnegative and normalized basis functions of T n presented in [13] and [14].…”

“…Many properties of the symmetric trigonometric basis functions can be found in [13] and [14]. In order to represent the trigonometric Hermite interpolants, we need to discuss the derivative properties of the basis functions.…”

“…, the interpolation operator T n reproduces all trigonometric polynomials of degree ≤ n in T n by (12), (13) and (17). (12), (13) andp [k] n,n .…”