Unification of the Bernstein-type polynomials and their applications
© Simsek; licensee Springer 2013
Received: 12 November 2012
Accepted: 27 February 2013
Published: 20 March 2013
In this paper, we investigate some new identities related to the unification of the Bernstein-type polynomials, Bernoulli polynomials, Euler numbers and Stirling numbers of the second kind. We also give some remarks and applications of the Bernstein-type polynomials related to solving high even-order differential equations by using the Bernstein-Galerkin method. We also give some applications on these polynomials and differential equations.
MSC:11B68, 12D10, 14F10, 26C05, 26C10, 30B40, 30C15, 42A38, 44A10.
KeywordsBernstein polynomials generating function Bezier curves Laplace transform functional equation high-order differential equations Bernstein-Galerkin method Bernoulli polynomials Bernoulli numbers Euler polynomials Euler numbers Genocchi polynomials Genocchi numbers Stirling numbers of the second kind
Generating functions play an important role in the investigation of various useful properties of the sequences and differential equations. These functions are also used to find many properties and formulas for the sequences. In , the author constructed certain generating functions for the unification of the classical Bernstein polynomials. Using these generating functions, the author derived several interesting and useful identities for these polynomials. The Bernstein polynomials have been defined by many different ways, for example, by q-series, by complex function and by many algorithms. The Bernstein polynomials are used in approximations of functions as well as in other fields such as smoothing in statistics, in numerical analysis, constructing the Bezier curves. The Bernstein polynomials are also used to solve differential equations.
According to Farouki , the Bernstein polynomial basis was introduced 100 years ago (Bernstein, 1912) as a means to constructively prove the ability of polynomials to approximate any continuous function, to any desired accuracy, over a prescribed interval. Their slow convergence rate and the lack of digital computers to efficiently construct them caused the Bernstein polynomials to lie dormant in the theory rather than practice of approximation for the better part of a century. The Bernstein coefficients of a polynomial provide valuable insight into its behavior over a given finite interval, yielding many useful properties and elegant algorithms that are now being increasingly adopted in other application domains.
An explicit formula of the polynomials is given by the following theorem .
The remainder of this study is organized as follows.
Section 2: We give many properties of the unification of the Bernstein-type polynomials: partition of unity, alternating sum, subdivision property. We also give many functional equations and differential equations of this generating function. Using these equations, many properties of the unification of the Bernstein-type polynomials can be found. Section 3: Integral representations of the unification of the Bernstein-type polynomials are given. Using these representations, we give an identity. Section 4: By using the Laplace transform, we find some identities of the unification of the Bernstein-type polynomials. Section 5: By using a new generating function, we prove the Marsden identity for the unification of the Bernstein-type polynomials. Section 6: By using generating functions, we give relations between the unification of the Bernstein-type polynomial, the unification of the Bernoulli polynomial of higher order and the Stirling numbers of the second kind. Section 7: By using the unification of the Bernstein-type polynomials and the Bernstein-Galerkin methods, we solve high even-order differential equations. Section 8: We give some remarks on the unification of the Bernstein-type polynomials and Bezier-type curves.
2 Properties of the unification of the Bernstein-type polynomials
In this section, we investigate some properties of the unification of the Bernstein-type polynomials.
2.1 Partition of unity
2.2 Alternating sum
By using the same method as that in  and (2), we arrive at a formula for the alternating sum of the polynomials , which is given by the following theorem.
2.3 Subdivision property
Here, we give partial differential equations and a functional equation of the generating function for the unification of the Bernstein-type polynomials . By using this functional equation, we derive the subdivision property unification of the Bernstein-type polynomials .
By using the above functional equation and (2), we derive the subdivision property for the polynomials by the following theorem.
where denotes the classical Bernstein basis function.
By applying these partial differential equations, we obtain the following derivative relations which are related to the subdivision property unification of the Bernstein-type polynomials , respectively:
3 Integral representations
In this section, we derive integral representations of the unification of the Bernstein-type polynomials . We also give an identity which connects the binomial coefficients, gamma and beta functions.
By using (5), we easily arrive at the desired result. □
In this section, by using the Laplace transform, we give some identities of the unification of the Bernstein-type polynomials .
by substituting (12) into (11), we arrive at the following theorem.
Remark 4.2 If we set in (13), then we arrive at Theorem 15 in .
Therefore, we arrive at the following theorem.
5 Marsden identity
In this section, by using generating functions, we prove the Marsden identity for the unification of the Bernstein-type polynomials . This identity is associated with a formula for rational linear transformation of B-splines, which are of interest in computer-aided geometric design and approximation theory.
Therefore, we arrive at the Marsden identity which is given by the following theorem.
6 Relations between the polynomial , unification of the Bernoulli polynomial of higher order and Stirling numbers of the second kind
The above generating function is related to some special polynomials as follows.
where denotes the classical Euler polynomials.
where and denote the unification Bernoulli polynomial of higher order and -Stirling numbers of the second kind, respectively.
From the above equation, after some calculation, we find the desired result. □
where denotes the Euler polynomials of higher order.
By using the Cauchy product in the above, after some calculation, we find the desired result. □
By substituting (20) into (19), we arrive at the following result.
7 Unification of the Bernstein-type polynomials for solving high even-order differential equations by the Bernstein-Galerkin methods
In , Doha et al. gave an application of the Bernstein polynomials for solving high even-order differential equations by using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods. The methods do not contain generating functions for proving explicitly the derivatives formula of the Bernstein polynomials of any degree and for any order in terms of Bernstein polynomials themselves. Here, we prove this formula for the unification of the Bernstein-type polynomials by a higher-order partial differential equation and functional equations. We also give some remarks and applications related to these polynomials and the Bernstein-Galerkin method.
By using the same method as in , we now give a higher-order partial differential equation for the generating function as follows.
By substituting (21) into the above equation, after some calculation, we arrive at the following theorem.
By applying the Bernstein-Galerkin approximation (24), we find as follows. For solving this equation, we need the following notations, which we recall from the work of Doha et al. [, p.9, Eq. (4.4)].
where for all . The 2m boundary conditions lead to the first m, and the least m expansion coefficients are zero.
8 Further remarks on Bezier curves
The unification of the Bernstein-type polynomials is used to construct Bezier-type curves which are used in computer-aided graphics design and related fields and also in the time domain, particularly in animation and interface design (cf. [2, 6]).
where , denotes the unification of the Bernstein-type polynomials and are the control points.
The unification of the Bernstein-type polynomials might affect the shape of the curves.
The author completed the paper himself. The author read and approved the final manuscript.
Dedicated to Professor Hari M Srivastava.
The present investigation was supported by the Scientific Research Project Administration of Akdeniz University.
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