Modified Stancu operators based on inverse Polya Eggenberger distribution

In this paper, we construct a sequence of modified Stancu-Baskakov operators for a real valued function bounded on [0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0,\infty)$\end{document}, based on a function τ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau(x)$\end{document}. This function τ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau(x)$\end{document} is infinite times continuously differentiable on [0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0,\infty)$\end{document} and satisfy the conditions τ(0)=0,τ′(x)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau (0)=0,~\tau ^{\prime}(x)>0$\end{document} and τ″(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau^{\prime\prime}(x)$\end{document} is bounded for all x∈[0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\in {}[0,\infty)$\end{document}. We study the degree of approximation of these operators by means of the Peetre K-functional and the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja-type theorems are also established in terms of the first order Ditzian-Totik modulus of smoothness.


Introduction
In , Eggenberger and Pólya [] were the first to introduce Pólya-Eggenberger distribution. After that, in , Johnson and Kotz [] gave a short discussion of Pólya-Eggenberger distribution.
Proof On differentiating μ n,m (x) with respect to x, the proof of the recurrence relation easily follows; hence the details are omitted.
Remark  From Lemma , we have The values of the Stancu-Baskakov operators (.) for the test functions e i (t) = t i , i = , , , are given in the following lemma.
Proof The identities (i)-(iii) are proved in [], hence we give the proof of the identity (iv). The identity (v) can be proved in a similar manner. We have Therefore using Remark , we get Now, by a simple calculation, we get the required result.
As a consequence of Lemma , we obtain the following.
Lemma  For the Stancu-Baskakov operator (.), the following equalities hold: Let  ≤ r n (x) ≤  be a sequence of continuous functions for each x ∈ [, ] and n ∈ N. Using this sequence r n (x), for any f ∈ C[, ], King [] proposed the following modification of the Bernstein polynomial for a better approximation: and studied global smoothness preservation, the approximation of decreasing and convex functions, the validity of a Voronovskaja-type theorem and a recursion formula generalizing a corresponding result for the classical Bernstein operators. Motivated by the above work, in the present paper we introduce modified Stancu-Baskakov operators based on a function τ (x) and obtain the rate of approximation of these operators with the help of Peetre's K-functional and the Ditzian-Totik modulus of smoothness. Also, we prove a quantitative Voronovskaja-type theorem by using the first order Ditzian-Totik modulus of smoothness.
Throughout this paper, we assume that C denotes a constant not necessarily the same at each occurence.

Modified Stancu-Baskakov operators
Lemma  The operator defined by (.) satisfies the following equalities: Proof The proof of lemma is straightforward on using Lemma . Hence we omit the details.
Let the mth order central moment for the operators given by (.) be defined as Lemma  For the central moment operator μ τ n,m (x), the following equalities hold: Proof Using the definition (.) of the modified Stancu-Baskakov operators and Lemma , the proof of the lemma easily follows. Hence, the details are omitted. Let For f ∈ C B [, ∞) and δ > , the Peetre K -functional [] is defined by where g W  = g + g + g .
From [], Proposition .., there exists a constant C >  independent of f and δ such that where ω  is the second order modulus of smoothness of f ∈ C B [, ∞) and is defined as In the following, we assume that inf x∈[,∞) τ (x) ≥ a, a ∈ R + := (, ∞). Next, we recall the definitions of the Ditzian-Totik first order modulus of smoothness and the K -functional []. Let φ τ (x) := √ τ (x)( + τ (x)) and f ∈ C B [, ∞). The first order modulus of smoothness is given by Further, the appropriate K -functional is defined by Proof By the definition of the modified Stancu-Baskakov operators (.) and using Lemma  we have for every x ∈ [, ∞). Hence the required result is immediate.
Theorem  Let f ∈ C B [, ∞). Then, for n ≥ , there exists a constant C >  such that on each compact subset of [, ∞).
Proof Let U be a compact subset of [, ∞). For each x ∈ U, first we define an auxiliary operator as Now, using Lemma , we have Let g ∈ W  , x ∈ U and t ∈ [, ∞). Then by Taylor's expansion, and using results in [], p., we get Now, applying the operator V *  n ,τ n (·; x) to both sides of the above equality, we get Again, for each x ∈ U, we have Now, using the definition of the auxiliary operators, Theorem  and inequality (.), for each x ∈ U we have Taking the infimum on the right side of the above inequality over all g ∈ W  and for all x ∈ U, we have using equation (.), we get the required result.
Theorem  Let f ∈ C B [, ∞). Then for every x ∈ [, ∞), and n ≥  we have Proof For any g ∈ W φ τ [, ∞), by Taylor's expansion, we have Applying the operator V  n ,τ n (·; x) on both sides of the above equality, we get