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Quasi-interpolant operators in Bernstein basis

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Abstract

The aim of this paper is to present a family of polynomial quasi-interpolants in Bernstein basis. More precisely, we will combine the strong features of the polar forms and the symmetric polynomials to derive the coefficients of the quasi-interpolant in the Bernstein basis representation. We also derive a collection of spline quasi-interpolants that reproduce polynomial functions up to degree 2. Numerical examples support the theoretical results and show that the proposed scheme is simple and effective.

Introduction

Bernstein polynomial bases are incredibly useful mathematical tools as they are simply defined, can be easily implemented on computer systems and represent a very large range of functions and curves (see [7]). Among the strongest properties of these polynomials, they possess the partition of unity one and are non-negative. There is a fair amount of literature on Bernstein polynomials [7]. Their roots are 0 or 1 with multiplicities and as stated in [20], Bernstein polynomials form a complete basis over the interval [0,1] where they can be produced recursively. These strong properties we mentioned and others, with the fact that the Bernstein basis is known to be optimally stable, even if the convergence of the Bernstein polynomials is slow, have compensating properties of shape preservation. See also [2], [7] for a recent survey of Bernstein polynomials from the historical perspective. This makes it a good choice to construct some discrete quasi-interpolants which are the main tool of this work. Various methods for building univariate Bernstein quasi-interpolant operators have been developed in the literature (see [21], and references therein). For example, P. Sablonnière [16] used the classical Bernstein operators or Durrmeyer–Derriennic operators for constructing some families of quasi-interpolants schemes. The Bernstein–Kantorovich operators have been used by Guo et al. [9] and Liu et al. [11] for building a family of quasi-interpolants with polynomial coefficients. The polar forms [10], [18] are used to study a huge variety of recursive schemes that involve polynomials and splines, partly because of its effectiveness and partly because of its elegant mathematical formulation. The expression of monomials in terms of Bernstein basis was already provided in [6]. The use of polar forms for the construction of approximation schemes in the Bernstein basis representation is not new. For instance, polar forms for the Bernstein basis (in the general multivariate setting) have been used by H. Speleers in [19] to compute the Hermite interpolation with polynomials and splines.

The purpose of this paper is to show how polar forms can also be used as a powerful tool for producing new quasi-interpolant operators in Bernstein basis. More precisely, the polar forms tool is used to introduce some discrete quasi-interpolants of the form Qf(x)=k=0nfk,nBk,n(x), which are exact on Pn.

We will compute the coefficients fk,n, by using numbers like Stirling numbers Si,j[n], which are strongly linked to the elementary symmetric polynomials [13] and the Stirling numbers of the first kind [1]. The use of special numbers (like the Stirling numbers) is also not new in the construction of spline quasi-interpolants. A nice review is given in [5] where the authors used the central factorial numbers (cfn) in spline and approximation theories.

The outline of this paper is as follows. In Section 2, we give a Marsden’s identity in Bernstein basis and we recall the definition and main properties of the polar forms that are needed in our proof. In Section 3, with the help of homogeneous polar forms, we construct some discrete quasi-interpolants which are exact for polynomials of degree n and give some explicit examples. Finally, the paper is concluded in Section 4.

Section snippets

Bernstein basis and polar forms

This section recalls some of the main properties of polar forms and gives a Marsden’s identity in Bernstein basis using polar forms. It also introduces the elementary symmetric polynomials, since they play an important part in the construction of the present quasi-interpolant operators.

The Bernstein polynomials of degree n on [0,1], also seen as special cases of B-splines are defined by : Bk,n(x)=nkxk(1x)nk,k=0,,nx[0,1].The internal nodes of the kth Bernstein polynomial are only formed by 0

Quasi-interpolant operators in Bernstein basis

The purpose of this section is to find the best possible form to represent some quasi-interpolants in Bernstein basis. For that, we give a new proof of the polynomial reproduction property of Bernstein basis. To do so, we combine the Lagrange polynomial approximation method with the symmetric functions. The resulting quasi-interpolant reproduces the polynomials exactly. We will also discuss, how to calculate recursively the Stirling numbers.

Conclusion

In this paper, some spline quasi-interpolants in Bernstein basis are presented. They are based on the new numbers which perfectly resemble to Stirling numbers. The performance of this method is confirmed by some numerical examples. Furthermore, the coefficients of the proposed quasi-interpolant schemes are simple and they can be easily calculated by using one of the algorithms presented in this document.

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