Open Access Research

Properties of q-analogue of Beta operator

Vijay Gupta1, Honey Sharma2, Taekyun Kim3* and Sang-Hun Lee4

Author Affiliations

1 School of Applied Science, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi-110078, India

2 Department of Mathematics, Dr. B. R. Ambedkar National Institute of Technology, Jalandhar-144011 (Punjab), India

3 Department of Mathematics, Kwangwoon University, Seoul 139-701, South Korea

4 Division of General Education, Kwangwoon University, Seoul 139-701, South Korea

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Advances in Difference Equations 2012, 2012:86  doi:10.1186/1687-1847-2012-86


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2012/1/86


Received:20 April 2012
Accepted:25 June 2012
Published:25 June 2012

© 2012 Gupta et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, we introduce the q-variant of Beta operator. We find the recurrence formula for mth-order moments. Here, we establish some direct theorems in terms of modulus of continuity for these operators. We also propose conditions for better approximation. In the end, we also propose the Stancu-type generalization.

Mathematical Subject Classification: 41A25; 41A35.

Keywords:
q-integers; q-beta function; q-Beta operators; modulus of continuity; asymptotic formula

1. Introduction

In the last four decades after the integral modification of Bernstein polynomials by Durrmeyer, several new Durrmeyer-type operators were introduced and their approximation properties were discussed. In 2007, Gupta et al. [1]proposed a family of linear positive operators as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M1">View MathML</a>

(1.1)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M2">View MathML</a>

for f Cγ [0, ), where Cγ [0, ), γ > 0 be the class of all continuous functions defined on [0, ) satisfying the growth condition |f (t)| Ctγ, C > 0 and B(k, n + 1) is beta function. They [1] established the direct and inverse results for these operators.

In the recent years, q-calculus was used in approximation theory and several new operators were introduced and their approximation properties were discussed (see [2-6], etc.). Motivated by these operators, we now introduce the q-analogue of (1.1). For f C[0,∞) and 0 < q < 1, we propose the q-Beta operators as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M3">View MathML</a>

(1.2)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M4">View MathML</a>

and Bq(t, s) denote the q-Beta function [7] is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M5">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M6">View MathML</a>. In particular, for any positive integer n, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M7">View MathML</a>, K(x,0) = 1, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M8">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M9">View MathML</a>, n ∈ ℕ, a, b ∈ ℝ.

This article is the extension of the earlier work of [1]. Here, we consider the q variant of the operators discussed in [1] and obtain the recurrence relations for moments. We also obtain some direct results for the q operators, which also include the asymptotic formula. In the end, we establish the conditions for better approximation.

2. Preliminaries

To make the article self-content, here we mention certain basic definitions of q-calculus, details can be found in [8,9] and the other recent articles. For each nonnegative integer k, the q-integer [k]q and the q-factorial [k]q! are, respectively, defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M10">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M11">View MathML</a>

For the integers n, k satisfying n k ≥ 0, the q-binomial coecients are defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M12">View MathML</a>

The q-derivative of a function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M13">View MathML</a> is defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M14">View MathML</a>

and the q-improper integral (see [10]) is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M15">View MathML</a>

The q-integral by parts is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M16">View MathML</a>

Lemma 1. For n, k ≥ 0, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M17">View MathML</a>

(2.1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M18">View MathML</a>

(2.2)

Proof. Using q-derivative operator, we can write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M19">View MathML</a>

Equation (2.2) can be obtained directly by using q-quotient rule as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M20">View MathML</a>

Remark 1. By using (2.1) and Dqxk−1= [k − 1]qxk−2, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M21">View MathML</a>

Hence, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M22">View MathML</a>

(2.3)

Lemma 2. We have following equalities

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M23">View MathML</a>

(2.4)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M24">View MathML</a>

(2.5)

Proof. Above equalities can be obtained by direct computations using definition of operator and (2.3). □

Theorem 1. If mth (m > 0, m ∈ ℕ)-order moment of operator (1.2) is defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M25">View MathML</a>

then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M26">View MathML</a> and for n > m, we have following recurrence relation,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M27">View MathML</a>

(2.6)

Proof. By using (2.4), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M28">View MathML</a>

by (2.5) and q-integration by parts, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M29">View MathML</a>

by combining above two equations, we can write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M30">View MathML</a>

Hence, the result follows. □

Corollary 2. We have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M31">View MathML</a>

(2.7)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M32">View MathML</a>

(2.8)

Corollary 3. If we denote central moments by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M33">View MathML</a>, m = 1,2, then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M34">View MathML</a>

(2.9)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M35">View MathML</a>

(2.10)

Remark 2. As a special case when q → 1-, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M36">View MathML</a>

The first two central moments for q → 1are

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M37">View MathML</a>

which are the moments obtained by Gupta et al. [1] for Beta operator.

3. Ordinary approximation

Let CB [0, ) be the space of all real valued continuous bounded function f on [0, ) endowed with the norm || f || = sup{|f(x)| : x ∈ [0, )}. Further let us consider the following K-functional:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M38">View MathML</a>

where δ > 0 and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M39">View MathML</a>

From [11], there exist an absolute constant C > 0 such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M40">View MathML</a>

(3.1)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M41">View MathML</a>

is the second-order modulus of smoothness of f CB [0, ). By

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M42">View MathML</a>

we denote the usual modulus of continuity of f CB [0, ).

Theorem 4. Let 0 < q < 1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M43">View MathML</a>

for every x ∈ [0, ) and f CB [0, ), where C is a positive constant.

Proof. We consider modified operators <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M44">View MathML</a> defined by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M45">View MathML</a>

(3.2)

x ∈ [0, ). The operators <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M46">View MathML</a> preserve the linear functions:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M47">View MathML</a>

(3.3)

Let g W2 and t ∈ [0, ). Using Taylor's expansion, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M48">View MathML</a>

and (3.3), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M49">View MathML</a>

Therefore, from (3.2), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M50">View MathML</a>

(3.4)

From Corollary 3, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M51">View MathML</a>

(3.5)

By (3.2) and Corollary 1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M52">View MathML</a>

(3.6)

Now using (3.2), (3.5), and (3.6), we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M53">View MathML</a>

Thus taking infimum on the right-hand side over all g W2, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M54">View MathML</a>

In the view of (3.1), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M55">View MathML</a>

This completes the proof of the theorem. □

4. Weighted approximation

Here, we give weighted approximation theorem for the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M56">View MathML</a>. Similar type of results are given in [3].

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M57">View MathML</a> be the set of all functions f defined on the interval [0, ) satisfying the condition

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M58">View MathML</a>

where Mf is a constant depending on f. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M59">View MathML</a> is a normed space with the norm

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M60">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M61">View MathML</a> denotes the subspace of all continuous functions in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M62">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M63">View MathML</a> denotes the subspace of all functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M64">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M65">View MathML</a>.

Theorem 5. Let q = qn ∈ (0, 1) such that qn 1 as n → ∞, then for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M66">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M67">View MathML</a>

(4.1)

Proof. By the Korovkin's theorem (see [12]), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M68">View MathML</a> converges to f uniformly as n → ∞ for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M69">View MathML</a> if it satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M70">View MathML</a> for i = 0, 1, 2 uniformly as n → ∞.

As, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M71">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M72">View MathML</a>

(4.2)

By Corollary 2, for n > 1,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M73">View MathML</a>

as n → ∞, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M74">View MathML</a>

(4.3)

Similarly for n > 1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M75">View MathML</a>

as n → ∞, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M76">View MathML</a>

(4.4)

By (4.2), (4.3), (4.4), and Korovkin's theorem, we get the desired result. □

Theorem 6. Let q = qn ∈ (0, 1) such that qn 1 as n → ∞, then for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M77">View MathML</a>and α> 0, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M78">View MathML</a>

(4.5)

Proof. For any fixed x0 > 0, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M79">View MathML</a>

By Theorem 5 and Corollary 2 first two terms of above inequality tends to 0 as n → ∞. Last term of inequality can be made small enough for large x0 > 0. This completes the proof.

5. Central moments and asymptotic formula

In this section, we observe that it is not possible to estimate recurrence formula <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M80">View MathML</a> in q calculus, there may be some techniques, but at the moment it can be considered as an open problem. Here we establish the recurrence relation for the central moments and obtain asymptotic formula.

Lemma 3. If we de ne the central moments as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M81">View MathML</a>

then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M82">View MathML</a>

and for n > m, we have the following recurrence relation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M83">View MathML</a>

proof. Using the identity

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M84">View MathML</a>

and q derivatives of product rule, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M85">View MathML</a>

Thus

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M86">View MathML</a>

Using the identities

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M87">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M88">View MathML</a>

we obtain the following identity after simple computation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M89">View MathML</a>

Using the above identity and q integral by parts

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M90">View MathML</a>

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M91">View MathML</a>

Finally, using

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M92">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M93">View MathML</a>

we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M94">View MathML</a>

This completes the proof of recurrence relation.

Theorem 7. Let f C [0, ) be a bounded function and (qn) denote a sequence such that 0 < qn < 1 and qn 1 as n → ∞. Then we have for a point x ∈ (0, )

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M95">View MathML</a>

Proof. By q-Taylor's formula [7] on f, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M96">View MathML</a>

for 0 < q < 1, where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M97">View MathML</a>

(5.1)

We know that for n large enough

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M98">View MathML</a>

(5.2)

That is for any ε > 0, A > 0, there exists a δ > 0 such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M99">View MathML</a>

(5.3)

for |t − x| < δ and n sufficiently large. Using (5.1), we can write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M100">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M101">View MathML</a>

We can easily see that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M102">View MathML</a>

In order to complete the proof of the theorem, it is suffficient to show that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M103">View MathML</a>. We proceed as follows:

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M104">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M105">View MathML</a>

so that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M106">View MathML</a>

where χx (t) is the characteristic function of the interval {t : |t − x| < δ}.

It follows from (5.1)

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M107">View MathML</a>

if |t − x| δ, then <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M108">View MathML</a>, where M > 0 is a constant. Since

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M109">View MathML</a>

we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M110">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M111">View MathML</a>

Using Lemma 3, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M112">View MathML</a>

Thus, for n suciently large <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M113">View MathML</a>. This completes the proof of theorem. □

Corollary 8. Let f C [0, ) be a bounded function and (qn) denote a sequence such that 0 < qn < 1 and qn 1 as n → ∞. Suppose that the first and second derivative f '(x)

and f '' (x) exist at a point x ∈ (0, ), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M114">View MathML</a>

6. Better error approximation

King [13] in 2003 proposed a new approach to modify the Bernstein polynomials to improve rate of convergence, by making operator to preserve test functions e0 and e1. As the q-Beta operators <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M115">View MathML</a> reproduce only constant functions, this motivated us to propose the modification of (1.2), so that they reproduce constant as well as linear functions.

Define sequence {un,q(x)} of real valued continues functions on [0, ) with 0 ≤ un,q(x) < ∞, as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M116">View MathML</a>

We replace x in definition of operator (1.2) with un,q(x). Therefore, modified operator is given as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M117">View MathML</a>

(6.1)

Remark 3. By simple computation we can write

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M118">View MathML</a>

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M119">View MathML</a>

Theorem 9. Let f CB(I), then for every x I and for n > 1, C > 0, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M120">View MathML</a>

(6.2)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M121">View MathML</a>.

Proof. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M122">View MathML</a>, by Taylor's series

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M123">View MathML</a>

therefore, by linearity and Remark 3, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M124">View MathML</a>

Also

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M125">View MathML</a>

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M126">View MathML</a>

on choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M127">View MathML</a>, taking infimum over <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M122">View MathML</a>, we get the desired result.

Theorem 10. Let q = qn satisfies 0 < qn < 1 and let qn 1 as n → ∞. For each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M128">View MathML</a>we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M129">View MathML</a>

Proof. To prove theorem (Using the Theorem in [14]), it is sufficient to verify the following three conditions

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M130">View MathML</a>

(6.3)

By Remark 3, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M131">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M132">View MathML</a>, the first and second condition of (6.3) is fulfilled for ν = 0 and ν = 1.

And

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M133">View MathML</a>

which implies that as n → ∞

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M134">View MathML</a>

Thus, the proof is completed. □

7. Stancu approach

In 1968, Stancu introduced Bernstein-Stancu operators in [15] a linear positive operator depending on two non-negative parameters α and β satisfying the condition 0 ≤ α β. Recently, many researcher applied this approach to many operators, for detail see [16,17,2], etc.

For f C[0, ) and 0 < q < 1, we define the q-Beta Stancu operators as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M135">View MathML</a>

(7.1)

where 0 ≤ α β.

In each of following theorems, we assume that q = qn, where qn is a sequence of real numbers such that 0 < qn < 1 for all n and limn→∞ qn = 1.

Theorem 11. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M136">View MathML</a>, s = 0, 1, 2 the following identities hold:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M137">View MathML</a>

Proof of the theorem can be obtained directly by using linearity of operator and Corollary 2.

Corollary 12. If we denote central moments by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M138">View MathML</a>, m = 1, 2, then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M139">View MathML</a>

Theorem 13. Let 0 < q < 1, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2012/1/86/mathml/M140">View MathML</a>

for every α, β ≥ 0, x ∈ [0, ) and f CB [0, ), where C is a positive constant.

Proof of theorem is just similar to Theorem 4.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

Acknowledgements

The authors would like to express their deep gratitude to the referees for their valuable suggestions and comments.

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