Abstract
The truncated -Bernstein polynomials , , and emerge naturally when the -Bernstein polynomials of functions vanishing in some neighbourhood of 0 are considered. In this paper, the convergence of the truncated -polynomials on is studied. To support the theoretical results, some numerical examples are provided.
1. Introduction
Let . For any nonnegative integer , the -integer is defined by and the -factorial is defined by For integers , the -binomial coefficient is defined by Clearly, for , We also use the following standard notations:
These notations and definitions can be seen in [1, Chapter 10]. The following generalization of the Bernstein polynomials based on the -integers has been introduced by Phillips in [2].
Definition 1. For any , the -Bernstein polynomials of are defined by where
Whereas one can obtain the classical Bernstein polynomials for , the name “-Bernstein polynomials” will be used to refer to the case . For more information and open problems related to the -Bernstein polynomials the reader is referred to [3].
The -Bernstein polynomials as well as the classical Bernstein polynomials have some common properties (cf. [4] and references therein), such as the end-point interpolation property, and the shape-preserving properties in the case . Similar to the Bernstein polynomials, the -Bernstein polynomials are degree reducing on the set of polynomials.
As for the convergence properties of the -Bernstein polynomials, they demonstrate a striking difference from those of the Bernstein polynomials. In general terms, this is true for various Bernstein-type operators based on the -integers, and it is exactly the occurrence of new phenomena and insights that makes the study of the convergence of these operators attractive and challenging. During the last decade, the approximation by operators based on the -integers has been investigated by many researchers (see, e.g., [5–10]).
In this paper, the convergence of the truncated -Bernstein polynomials is addressed. Such polynomials come into the picture once the -Bernstein polynomials are considered for the functions vanishing in some neighbourhood of 0. The truncation of functions and operators is a well-known and widely used tool in functional analysis and approximation theory. See, for example, a recent paper [11] where it has been used to prove a Daugavet-type inequality. As for the truncation of the classical Bernstein polynomials, it has been used by Cooper and Waldron in [12].
Definition 2 (see [13]). For any , , and , the -truncated -Bernstein polynomials of are
In this paper, the properties of polynomials (9) are studied only in the case . Throughout the paper, and unless stated otherwise, it is assumed that is fixed. For the sake of simplicity, the notations and for will be used.
For , , the function will be called the -truncation of . Clearly, if and only if for .
The paper is organized as follows. The next section is devoted to the main results, which, finally, are illustrated in Section 3 using numerical examples.
2. Results
In this section, the results are presented pertaining the convergence of the truncated -Bernstein polynomials. The following simple assertion reveals that the truncated -Bernstein polynomials appear naturally when we consider those functions vanishing in some neighbourhood of 0.
Theorem 3. If , then , where is given by (11).
Proof. Since , for , one has while The right sides of (12) and (13) are identical if and only if The inequality on the right holds for all , while the one on the left is true if and only if .
Remark 4. If , then for all .
Corollary 5. If for , then .
Theorem 2.1 of [7] states that, for any , , and , . That is, when , the sequence of polynomials converges to on . The following lemma, which can be inferred from Lemma 3.1 of [13], is needed to investigate the convergence of the truncated -Bernstein polynomials.
Lemma 6. Let . Then where .
Note that if and only if . Thus, the following holds.
Lemma 7. For any and , . Moreover, the convergence is uniform on any compact subset of .
Proof. Since is continuous on , there exists such that for all . In addition, from the definition of , one can obtain for . Hence, for any , where . This yields the desired result.
Clearly, for , the truncated -Bernstein polynomials have the form whence
As the next theorem reveals, the case is not straightforward.
Theorem 8. Let , . If , then
Proof. The assertion (20) is obvious by Lemma 7 and the convergence on . To see (21), let . Then, Lemma 6 implies that Moreover, it is easy to see that as . Therefore, Since and , the statement follows.
In the case when , the truncated -Bernstein polynomials behave differently as the following theorem indicates.
Theorem 9. Let with , and suppose that, for each , there exists such that Setting , one has the following: (i)if , then except possibly at most at points, outside of ;(ii)if , then for all .
Proof. From (24), for each , one has as , which together with Lemma 6 and the fact that yields that where . Now, suppose that takes on its maximum value at with . Then, Since is a polynomial of degree at most , can vanish at most at points. This completes the proof.
3. Numerical Examples
In this part, some numerical examples are given to demonstrate the theoretical results.
Example 10. Let , . Consider the function , defined by Since for and , by Theorem 8, In Figure 1, the graphs of and on some different intervals are given.
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Example 11. Let , , and consider a function satisfying and on the rest of , is piecewise linear. It is easily seen that the conditions of Theorem 9 are fulfilled by , , , , , and . Here, and vanishes at , , and . Therefore, In Figure 2, the graphs of and on some different intervals are provided.
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Example 12. Let , , and consider the function with At any other point in , is defined in such a way that it is continuous. Obviously, the conditions of Theorem 9 are met by , , , , , , , , , and . Now, and . Therefore, In Figure 3, the graphs of and on some different intervals are depicted.
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Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author is grateful to the anonymous referees for their valuable comments and criticism. Also, the author would like to express appreciation to Mr. P. Danesh, Atilim University Academic Writing and Advisory Centre, for his help in the improvement of the presentation of the paper.