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.
1. Introduction
A century ago Bernstein [1] introduced his famous polynomials by defining where is a function defined on the interval and is a positive integer. As Bernstein proved, if is continuous on the interval then its sequence of Bernstein polynomials converges uniformly to on . 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 [4–6]). 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 [10–13]). 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 [16–19]). One may use the cosine polynomial sequence 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 . 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 polynomials is described and its convergence is discussed. Trigonometric quasi-interpolants of reproducing one degree of trigonometric polynomials are given in Section 4.
2. Trigonometric Basis Functions
Definition 1. For , , let , , ; one defines trigonometric polynomials of degree as follows: where
We choose domain in Definition 1 so that and are monotone, and is convex. From (3) and (4), we can obtain the coefficients of the trigonometric polynomials as Table 1.
Property 2. Linear independence property: the set of the trigonometric polynomials is linearly independent on .
Proof. Consider the trigonometric polynomial space
we know that
and then
On the other hand,
and ; we have
Hence,
Since the set of the trigonometric polynomials is linearly independent, we conclude that the set of the trigonometric polynomials is linearly independent on .
The set of the trigonometric polynomials forms a basis for the trigonometric polynomial space . We refer to the trigonometric functions as trigonometric basis functions.
Figure 1 shows the graphs of trigonometric basis functions with on the left and with on the right.
Now we show that trigonometric sequence has different properties than the sequence . Some important properties of the following are useful in the interest of constructing trigonometric polynomial approximants.
Property 3. Positivity of the basis functions: if , then , .
Proof. From (3) and (4), it is easy to see that for all possible . Since , it follows that .
Property 4. Partition of unity for the basis functions: for all , we have
Proof. Obviously, We assume that the formula is true for . Since for or , from (2), (4), and we have This is (11) with replaced by ; the proof is complete.
Property 5. Symmetry of the basis functions: for , we have
Proof. Obviously, . Assume ; from (4) we have for . These imply that the coefficients of are symmetric. Thus, for , we have
Based on Property 5, we refer to the basis functions as symmetric trigonometric basis functions.
Property 6. Recurrence relation of the basis functions: for and , we have where .
Proof. From (2) and (4), for , we have For , we have
Property 7. Degree elevation: for all , we have for , for .
Proof. For , by (2) we have From this we obtain (20). In the same way, we have (21).
Property 8. Derivative of the basis functions: for , we have For , we have
Proof. For , we have This implies the case of (23). For , we have This implies the cases of (23). In the same way, we can obtain the results on the other cases.
Property 9. Maximum values: for , obtains its maximum value at
Proof. Directly derivation computing to (2), we have for , and for . Since , we obtain for , and for . Let ; we have and then From this we obtain (28).
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 10. Explicit formula: for the coefficients of the trigonometric basis functions given by (4), we have
Proof. Since , , , (34) holds for and obviously. We assume that the formula (34) is true for , ; then By (4), for even numbers , we have For odd numbers , we have By induction, the proof is complete.
Property 11. Recurrence relation of the coefficients: for the coefficients of the trigonometric basis functions given by (4), we have
Proof. For , 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 . By (4), we have , . When is an odd number, we have When is an even number, analogously, we have
By Property 10 or Property 11, we have and so on.
Property 12. Positivity of the coefficients: for ,
Proof. Obviously, (43) holds when . For , by (38) we have Then, for , These equalities also hold for . When , By induction and symmetry, (43) holds.
3. Symmetric Trigonometric Polynomials
3.1. The Construction of the Trigonometric Polynomials
We will discuss trigonometric polynomial approximation on the special interval because the change of variable can be used to go back and forth between and .
Definition 13. Given nodes , and function values , we define trigonometric polynomials as follows:
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 , then .
For computing conveniently, we can choose nodes . On the convergence of , two kinds of the nodes will be discussed. One kind of the nodes is , , and for . Another kind of the nodes is , , , and for , where
We can also rewrite
By Property 11, expression (48) can be changed to and (49) can be changed to for . By Property 12, we have Therefore, for , it is easy to show that the node sequences (48) and (49) are monotonely increasing, respectively. In the following section, we can see that for (48) or (49).
Example 14. Let us consider the function as follows: Figure 2 shows the approximation curves of this function. On the left of Figure 2, 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 2, 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.
Example 15. Let us consider the function as follows: Figure 3 shows the approximation curves of this function. 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.
3.2. The Convergence of the Trigonometric Polynomials
The following theorem will be used repeatedly for the proof of the convergence of the trigonometric polynomials.
Theorem 16. For the coefficients of trigonometric basis functions, one has
Proof. Obviously, (57) holds for . For , by (38),
we obtain
Then, by (44),
Since
we have
From this we obtain (57).
Let ; we have
and then, by recursion,
From (54) we have
Obviously, , ; thus we can deduce that and then is a monotone bounded sequence. Therefore, exists. From (64) we obtain
and then
From the proof of Theorem 16, we can see . From this and (54), it is easy to show for (48) or (49).
Property 4 implies that . In order to show the convergence of trigonometric polynomials , we need to discuss and .
By , we have
The node expression (48) is set in the light of (69).
Theorem 17. For the node expression (48), and converge uniformly to and , respectively, for .
Proof. By (54) we have and then . From this we have Therefore, and then In the same way, we have From (57) and (58), it is easy to see that and converge uniformly to and , respectively, for .
From the monotonicity of , we can know that Hence, with nodes (48), and for .
Based on (69), if we minimize for , then the results are (53). If we minimize for , then For (53) and (77), it is easy to validate that
The node expression (49) is set in the light of the results of the minimality.
Theorem 18. For nodes (49), and converge uniformly to and , respectively, for .
Proof. From (69), we have where Obviously, Therefore, by (57) and (58), and converge uniformly to and , respectively, for .
Obviously, with nodes (49), and for .
In the following, for the sake of simplicity, we set if it does not make a confusion.
Theorem 19. With nodes (48) or (49), the sequence of trigonometric polynomials converges uniformly to for all .
Proof. The proof is similar to the one used in proving Korovkin theorem; see [3, 20]. Let ; 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 ,
Let ; we have
Thus, for all and in , we have
and then
Since , then
thus the sequence converges uniformly to 0. Therefore, we can select so that
whenever . Then .
Theorem 20. Let . If , then where
Proof. It is easy to validate that Let ; then where is the truncated power function. Since is of one sign for , and is of one sign for , we have for some and . Therefore
3.3. The Convergence of the Derivative Functions
For the trigonometric polynomial (47), obviously, By Property 8, we obtain where for , for . By Property 11, we have for , and for .
Theorem 21. With node expression (49), if , then the sequence of trigonometric polynomials converges uniformly to for all .
Proof. Let
for , we have
Therefore, based on Theorem 19 and symmetry, we need only to show that and when for .
It is easy to show
for , and
for . Thus, we have
for some ,
for some , , , and
for some , .
Obviously, when ,
For , let
where
Since
using (59), we have
and then, using (59),
From these we obtain
Since and are bounded and
we can conclude that when .
By (44), we have
and then
This implies that when .
Using (44) repeatedly, we have
Thus we have
Therefore,
4. Quasi-Interpolation by the Trigonometric Polynomials
Like Bernstein polynomials, the convergence of the given trigonometric polynomials is slow. For reproducing one degree of trigonometric polynomials, we consider the following quasi-interpolant: where .
Based on (69), in order to reproduce one degree of trigonometric polynomials by (122), we need to choose and to satisfy the following equalities: for , for .
Solving the above equations, for , we have
Using the node expression (49), we obtain for , and for .
In the same way, let then (126) also holds for . Thus, with all the coefficients and , the quasi-interpolant (122) reproduces one degree of trigonometric polynomials.
Example 22. Let us consider the quasi-interpolant (122) for the functions and . For , we have On the left of Figure 4, for the function , the functional curve (dotted line), the quadratic trigonometric curve with node expression (49) (solid line), and the quasi-interpolation curve of quadratic trigonometric polynomial (dashed line) are shown, respectively. On the right of Figure 4, for the function , the functional curve (dotted line), the quadratic trigonometric curve with node expression (49) (solid line), and the quasi-interpolation curve of quadratic trigonometric polynomial (dashed line) are shown, respectively. Obviously, the quasi-interpolation of trigonometric polynomial approximates the functions better than the trigonometric polynomial does.
Theorem 23. With node expression (49) and the coefficients (126) for , if , then the sequence of the quasi-interpolation trigonometric polynomials converges uniformly to for all .
Proof. Obviously, By (104) and (105), we have for some . From this we get Since we have
Based on the reproducing property of , we can give an error expression. By we have where
5. 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 then the sequence of the trigonometric polynomials converges uniformly to the function on . 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.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author thanks the referees for their valuable advice. This research is supported by the National Natural Science Foundation of China (nos. 11271376 and 60970097).