The Trigonometric Polynomial Like Bernstein Polynomial

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.


Introduction
A century ago Bernstein [1] introduced his famous polynomials by defining where is a function defined on the interval [0, 1] and is a positive integer. As Bernstein proved, if is continuous on the interval [0, 1] then its sequence of Bernstein polynomials converges uniformly to on [0, 1]. Thus Bernstein polynomials are important because a constructive proof of Weierstrass' theorem is given. Later, because the Bernstein polynomials are shape preserving, they were found to have practical applications. Many generalizations of them have been proposed. Very fine brief accounts of the Bernstein polynomials are given in Davis [2] and Phillips [3]. However, there are few results on the constructive proof of trigonometric polynomial sequence approximating continuous function. Some authors are interested in the problem of constructing nonnegative trigonometric polynomials (see [4][5][6]). Trigonometric interpolation has been considered by Salzer [7] and Henrici [8]. Several other authors have addressed Hermite problems, even for arbitrary points. They were mostly interested in existence questions [9], convergence results, and formulae other than Lagrange's (see [10][11][12][13]). Quasi-interpolant on trigonometric splines has been discussed in [14]. In [15], authors approximate continuous functions defined on a compact set ∈ [− , ] by trigonometric polynomials. Some problems of geometric modeling are solved better by trigonometric splines. Some types of trigonometric splines have been introduced having different features (see [16][17][18][19]). One may use the cosine polynomial sequence {cos } ( = 0, 1, . . . , ) to approximate a continuous function, but this sequence is not a basis of the trigonometric polynomial space of order .
The purpose of this paper is to construct an explicit sequence of trigonometric polynomials like Bernstein polynomials. Thus, trigonometric polynomials may be used like Bernstein polynomials. It is well known that Bernstein polynomials have many applications and are appropriate for numerical computation. New trigonometric polynomials like Bernstein polynomials provide different expressions for function approximation. We will present a symmetric trigonometric polynomial basis of order and show how it works. Although one can construct trigonometric polynomials via simple ways, via trigonometric kernels, for example, we will construct simpler and more evident trigonometric polynomial which converges uniformly to a continuous function defined on the interval [0, /2]. The problem of reproducing one degree of trigonometric polynomials by trigonometric quasi-interpolants is also solved.
The remainder of this paper is organized as follows. In Section 2, the basis functions of the trigonometric polynomial space are presented and the properties of the basis functions are shown. In Section 3, a sequence of trigonometric 2 The Scientific World Journal polynomials is described and its convergence is discussed. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are given in Section 4.
The Scientific World Journal 5 In the proof of Property 5, we have shown that the coefficients of the trigonometric basis functions are symmetric. Now we give further properties of the coefficients of the trigonometric basis functions.
Property 11. Recurrence relation of the coefficients: for the coefficients of the trigonometric basis functions given by (4), we have Proof. For = + 1, + 2, . . . , 2 − 1, by the symmetry of the coefficients shown in the proof of Property 4, we can obtain (39) from (38). Therefore, we consider only the cases = 1, 2, . . . , − 1. By (4), we have 0, = 1, 1, = . When is an odd number, we have 6 The Scientific World Journal When is an even number, analogously, we have By Property 10 or Property 11, we have 0, = 1, for ≥ 5, and so on.

Symmetric Trigonometric Polynomials
Since the symmetry of the Trigonometric basis functions, we call (47) as symmetric trigonometric polynomials.
Obviously, is a linear operator. Based on Property 3, another property of these operator is that they are positive. This implies that if ≥ 0, then ( , ) ≥ 0.
On the left of Figure 3, the functional curve (dotted line), the quadratic trigonometric curve (solid line), the quartic Bernstein polynomial curve (dashed line), and the quartic trigonometric curve (dashdot line) are shown with equidistant nodes, respectively. On the right of Figure 3, the functional curve (dotted line), the quadratic trigonometric curve (solid line), the cubic trigonometric curve (dashed line), and the quartic trigonometric curve (dashdot line) are shown with node expression (48), respectively.

The Convergence of the Trigonometric Polynomials.
The following theorem will be used repeatedly for the proof of the convergence of the trigonometric polynomials.
In the following, for the sake of simplicity, we set = , if it does not make a confusion. Proof. The proof is similar to the one used in proving Korovkin theorem; see [3,20]. Let > 0; we want to prove that an integer exists such that where Since is continuous on a compact interval, it is uniformly continuous. Consequently, a positive exists such that for all and in By Property 8, we obtain where [1] for = 0, 1, . . . , − 1, Proof. Let [1] ( ) = , ( ) + , Therefore, based on Theorem 19 and symmetry, we need only to show that , → 1 and , → 0 when → ∞ for = 0, 1, . . . , .
Using (44) repeatedly, we have Thus we have By (104) Based on the reproducing property of ( , ), we can give an error expression. By (136)

Conclusion
A symmetric basis of trigonometric polynomial space and its some interesting properties are presented. Using the positive trigonometric basis, symmetric trigonometric polynomial approximants are constructed. The trigonometric polynomial is simple and evident and easy for numerical computing. We are also interested in the particular basis and the trigonometric polynomial approximants because a constructive proof of trigonometric polynomial sequence approximating continuous function is given. The trigonometric polynomials have similar properties to Bernstein polynomials. Two kinds of node sequences are chosen particularly to show the convergence. We show that if a function is continuous on the interval [0, /2] then the sequence of the trigonometric polynomials converges uniformly to the function on [0, /2]. The derivative sequence of the trigonometric polynomials is also convergent if the function is twice differentiable. The trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are constructed and the sequence is uniform convergent.