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Representation of \((p,q)\)-Bernstein polynomials in terms of \((p,q)\)-Jacobi polynomials
Journal of Inequalities and Applications volume 2017, Article number: 167 (2017)
Abstract
A representation of \((p,q)\)-Bernstein polynomials in terms of \((p,q)\)-Jacobi polynomials is obtained.
1 Introduction
Classical univariate Bernstein polynomials were introduced by Bernstein in a constructive proof for the Stone-Weierstrass approximation theorem [1], and they are defined as [2]
They form a basis of polynomials and satisfy a number of important properties as non-negativity (\(b_{i}^{n}(x) \geq0\) for \(0 \leq x \leq 1\)), partition of unity (\(\sum_{i=0}^{n} b_{i}^{n}(x)=1\)) or symmetry (\(b_{i}^{n}(x)=b_{n-i}^{n}(1-x)\)).
For a given real-valued defined and bounded function f on the interval \([0,1]\), the nth Bernstein polynomial for f is
Then, for each point x of continuity of f, we have \(B_{n}(f)(x) \to f(x)\) as \(n \to\infty\). Moreover, if f is continuous on \([0,1]\) then \(B_{n}(f)\) converges uniformly to f as \(n \to\infty\). Also, for each point x of differentiability of f, we have \(B_{n}'(f)(x) \to f'(x)\) as \(n \to\infty\) and if f is continuously differentiable on \([0,1]\) then \(B_{n}'(f)\) converges to \(f'\) uniformly as \(n \to \infty\).
Bernstein polynomials have been generalized in the framework of q-calculus. More precisely, Lupaş [3] initiated the application of q-calculus in area of the approximation theory, and introduced the q-Bernstein polynomials. Later on, Philips [4] proposed and studied other q-Bernstein polynomials. In both the classical case and in its q-analogs, expansions of Bernstein polynomials have been obtained in terms of appropriate orthogonal bases [5, 6].
Mursaleen et al. [7] recently introduced first the concept of \((p, q)\)-calculus in approximation theory and studied the \((p, q)\)-analog of Bernstein operators. The approximation properties for these operators based on Korovkin’s theorem and some direct theorems were considered [8]. Also, many well-known approximation operators have been introduced using these techniques, such as Bleimann-Butzer-Hahn operators [9] and Szász-Mirakyan operators [10]. Very recently Milovanović et al. [11] considered a \((p, q)\)-analog of the beta operators and using it proposed an integral modification of the generalized Bernstein polynomials. \((p,q)\)-analogs of classical orthogonal polynomials have been characterized in [12].
The main aim of this work is to obtain a representation of \((p,q)\)-Bernstein polynomials in terms of suitable \((p,q)\)-orthogonal polynomials, where the connection coefficients are proved to satisfy a three-term recurrence relation. For this purpose, we have divided the work in two sections. First, we present the basic definitions and notations. Later, in Section 3 we obtain the main results of this work relating \((p,q)\)-Bernstein polynomials and \((p,q)\)-Jacobi orthogonal polynomials.
2 Basic definitions and notations
Next, we summarize the basic definitions and results which can be found in [13–18] and the references therein.
The \((p,q)\)-power is defined as
The \((p,q)\)-hypergeometric series is defined as
where
and \(r, s \in \mathbb {Z}_{+}\) and \(a_{1p},a_{1q},\dots ,a_{rp},a_{rq},b_{1p},b_{1q},\dots,b_{sp},b_{sq},z \in \mathbb {C}\).
The \((p,q)\)-difference operator is defined as (see e.g. [14])
where the shift operator is defined by
and \((\text{${\mathcal {D}}_{p,q}$}f)(0)=f'(0)\), provided that f is differentiable at 0.
The \((p,q)\)-Bernstein polynomials are defined as
and can be expanded in the basis \(\{x^{k}\}_{k \geq0}\) as
From the definition of \((p,q)\)-Bernstein polynomials it is possible to derive the basic properties of \((p,q)\)-Bernstein polynomials.
-
(1)
Partition of unity
$$\sum_{i=0}^{n} b_{i}^{n}(x;p,q)=1. $$ -
(2)
End-point properties
$$b_{i}^{n}(0;p,q)= \textstyle\begin{cases} 1, & i=0, \\ 0, & \text{otherwise}, \end{cases}\displaystyle \qquad b_{i}^{n}(1;p,q)= \textstyle\begin{cases} 1, & i=n, \\ 0, & \text{otherwise}. \end{cases} $$
The \((p,q)\)-Jacobi polynomials are defined by
and they satisfy the second order \((p,q)\)-difference equation
The \((p,q)\)-Jacobi polynomials satisfy the three-term recurrence relation
where
and
3 Representation of \((p,q)\)-Bernstein polynomials in terms of \((p,q)\)-Jacobi polynomials
Lemma 3.1
The \((p,q)\)-Bernstein polynomials satisfy the following first order \((p,q)\)-difference equation:
Proof
The result can be obtained by equating the coefficients in \(x^{j}\). □
If we introduce the first order \((p,q)\)-difference operator
then
Lemma 3.2
The \((p,q)\)-Jacobi polynomials satisfy the following structure relation:
where
Proof
The result follows from (7) by equating the coefficients in \(x^{j}\). □
Theorem 3.1
The \((p,q)\)-Bernstein polynomials defined in (5) have the following representation in terms of \((p,q)\)-Jacobi polynomials defined in (7):
where the connection coefficients \(H_{k}(i,n;\alpha,\beta;p,q)\) satisfy the following three-term recurrence relation:
valid for \(1 \leq k \leq n-1\) with initial conditions
and
Proof
In order to obtain the result we shall apply the so-called Navima algorithm (see e.g. [19, 20] and the references therein) for solving connection problems. If we apply the first order linear operator \(L_{i,n}\) defined in (12) to both sides of (14) we have
From the three-term recurrence relation for \((p,q)\)-Jacobi polynomials it yields
Therefore, by using the structure relation for \((p,q)\)-Jacobi polynomials (13) we have
where \(\Lambda_{i}(k,i,n;\alpha,\beta;p,q)\) are given in (18).
As a consequence,
By using the linear independence of \(\{P_{k}(p^{3}x;\alpha,\beta ;p,q)\}\) we obtain the three-term recurrence relation (15) for the connection coefficients \(H_{k}(i,n;\alpha,\beta;p,q)\), where the initial conditions are obtained by equating the highest power in \(x^{k}\). □
4 Conclusions
In this work we have obtained a three-term recurrence relation for the coefficients in the expansion of \((p,q)\)-Bernstein polynomials in terms of \((p,q)\)-Jacobi polynomials. For our purposes some auxiliary results both for \((p,q)\)-Bernstein polynomials and \((p,q)\)-Jacobi polynomials have been derived.
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Acknowledgements
The authors thank both reviewers for their comments. This work has been partially supported by the Ministerio de Ciencia e Innovación of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER, and Xunta de Galicia, grants GRC 2015-004 and R 2016/022. The first author thanks the hospitality of Departamento de Estatística, Análise Matemática e Optimización of Universidade de Santiago de Compostela, and Departamento de Matemática Aplicada II of Universidade de Vigo during her visit.
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Each of the authors, FS, IA, MMJ, and JJN contributed to each part of this study equally and read and approved the final version of the manuscript.
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Soleyman, F., Area, I., Masjed-Jamei, M. et al. Representation of \((p,q)\)-Bernstein polynomials in terms of \((p,q)\)-Jacobi polynomials. J Inequal Appl 2017, 167 (2017). https://doi.org/10.1186/s13660-017-1443-7
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DOI: https://doi.org/10.1186/s13660-017-1443-7