On generalized integral Bernstein operators based on q-integers
Introduction
In 1885, Weierstrass proved Weierstrass approximation theorem: ‘Every continuous function on , can be uniformly approximated by a sequence of polynomial functions’.
In 1912, Bernstein gave an alternative proof of the Weierstrass’s theorem using the Bernstein polynomials. Since Bernstein polynomial plays an important role in approximation theory, many researchers have been studied Bernstein polynomial and its different generalizations [1], [2], [3], [4].
In recent years, due to the intensive generalizations of q-Calculus and its applications in many fields such as mathematics, physics and computer science, generalizations of classical operators related to q-Calculus have emerged. Since last decades, q-calculus has been studied rigorously because its latent application in Mathematics, Mechanics and Physics. After Philips [5] introduced and studied q analog of Bernstein polynomials, the applications of q-calculus in the approximation theory has become one of the major areas of research. Sofia and Ankara [6] discussed that, the q-Bernstein polynomials inherit some properties of the classical Bernstein polynomials. Among those properties they mention the end-point interpolation property, the shape-preserving properties in the case . Like the classical Bernstein polynomials, the q-Bernstein polynomials reproduce linear functions, and they are degree-reducing on the set of polynomials. Alternatively, the examination of the convergence properties of the q-Bernstein polynomials reveals that these properties are essentially different from those of the classical ones. Since, q-generalization of some positive operators have attracted much interest, and a great number of interesting results related to these operators have been obtained [7], [9], [11], [12], [10], [13], [8], [14], [15].
To approximate Lebesgue integrable functions on the interval , Durrmeyer introduced the integral modification of the well known Bernstein polynomials. In 1981 Derriennic [4] first studied these operators in details, q-generalization of this operators introduced by Gupta and Heping [16]. In the present article we study convergence behavior of the q-analog of the Stancu generalization of Durrmeyer operators, which was introduced by Mishra and Patel [17].
We recall some concept of q-calculus. All of the results can be found in [18], [19]. In what follows, q is a real number satisfying .
For ,andAlso for any real number , we have . In special case when is a whole number this definition coincides with the above definition.
The q-analog of integration, discovered by Thomae [20] in the interval is defined byFor any , the q-Beta function is defined asIn particular, for , we have
Section snippets
Construction of operators
We setAfter development of quantum calculus, Phillips [5] proposed the following q-Bernstein polynomials, which for each positive integer n and are defined asIn 2005, Derriennic [21] introduced a q-analog of the integral summation operators and established some approximation properties of q-Durrmeyer operators. Recently, Gupta [22] studied following q-analog of Durrmeyer operators and discussed
Estimation of moments
In this section we shall obtain .
Note that for and by the definition of q-Beta function (see [19]), we haveand Lemma 1 We have Lemma 2 We haveand[22]
Convergence of q-Durrmeyer–Stancu operators
Theorem 1 Let . Then the sequence converges to f uniformly on for each if and only if . Proof Since the operator are positive linear operators on by well known theorem due to Korovkin, it implies that as for if and only ifIf , then and for , hence (4.1) follows from Lemma 2. On the other hand, if we assume that for any
Statistical convergence
A sequence is said to be statistically convergent to a number L, denoted by , if for every whereis the natural density of set and is the characteristic function of K. For instantseries converges statistically, but does not exist. We note that every convergent sequence is a statistical convergent, but converse need not be true (details can be found in [38]).
As an application of the
Voronovskaja type theorem
In this section we establish a Voronovskaja type asymptotic formula for the operators : Lemma 4 Assume that as . Then, for every , we have Theorem 6 Let f be bounded and integrable on the interval , second derivative of f exists at a fixed point and such that as , then Proof Using Taylor’s expansion of f, we can write
Better estimation
To make the convergence faster, King [40] proposed an approach to modify the Bernstein polynomials, so that sequence preserve test functions and . As the operators reproduce only constant functions, this motivated us to propose the modification of the operators (2.1), so that they reproduce constant as well as linear functions, for this purpose we propose the modification of the operators (2.1) aswhere
Conclusion
The results of our lemmas and theorems are more general rather than the results of any other previous proved lemmas and theorems, which will be enrich the literate of Applications of quantum calculus in operator theory and convergence estimates in the theory of approximations by linear operators. The researchers and professionals working or intend to work in areas of analysis and its applications will find this research article to be quite useful. Consequently, the results so established may be
Acknowledgements
The authors would like to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research article. Special thanks are due to Prof. Theodore E. Simos, Editor in chief and Prof. Melvin R. Scott, Emeritus Editor of Applied Mathematics and Computation, for kind cooperation, smooth behavior during communication and for their efforts to send the reports of the
References (41)
A generalization of the Bernstein polynomials
J. Math. Anal. Appl.
(1997)- et al.
Convergence of generalized Bernstein polynomials
J. Approx. Theory
(2002) Sur approximation de fonctions integrables sur par des polynomes de Bernstein modifies
J. Approx. Theory
(1981)Properties of convergence for -Bernstein polynomials
J. Math. Anal. Appl.
(2008)- et al.
Weighted statistical convergence and its application to Korovkin type approximation theorem
Appl. Math. Comput.
(2012) - et al.
A note on approximation properties of q-Durrmeyer operators
Appl. Math. Comput.
(2010) Some approximation properties of q-Durrmeyer operators
Appl. Math. Comput.
(2008)Approximation by Stancu–Chlodowsky polynomials
Comput. Math. Appl.
(2010)- et al.
On Stancu type generalization of q Baskakov operators
Math. Comput. Model.
(2010) - et al.
On the trigonometric approximation of signals belonging to generalized weighted Lipschitz -class by matrix operator of conjugate series of its fourier series
Appl. Math. Comput.
(2014)