© Hindawi Publishing Corp. RATE OF CONVERGENCE OF BOUNDED VARIATION FUNCTIONS BY A BÉZIER-DURRMEYER VARIANT OF THE BASKAKOV OPERATORS

We consider a Bezier-Durrmeyer integral variant of the 
Baskakov operators and study the rate of convergence for 
functions of bounded variation.


Introduction.
Let W (0, ∞) be the class of functions f which are locally integrable on (0, ∞) and are of polynomial growth as t → ∞, that is, for some positive r , there holds f (t) = O(t r ) as t → ∞.The Durrmeyer variant V n of the Baskakov operators associates to each function f ∈ W (0, ∞) the series where is the Baskakov basis function.Note that (1.1) is well defined, for n ≥ r + 2, provided that f (t) = O(t r ) as t → ∞.The operators (1.1) were first introduced by Sahai and Prasad [9].They termed these operators as modified Lupaş operators.In 1991, Sinha et al. [10] improved and corrected the results of [9] and denoted V n as modified Baskakov operators.The rate of convergence of the operators (1.1) on functions of bounded variation was studied in [8,11].We mention that Agrawal and Thamer [2] considered the variant of the operators (1.1) and studied its properties in subsequent papers [3,4,5].See also [1].The rate of convergence of the operators discussed by Agrawal and Thamer was studied by the first author in [7].
For each function f ∈ W (0, ∞) and α ≥ 1, we consider the Bézier-type Baskakov-Durrmeyer operators V n,α as where It is obvious that V n,α are positive linear operators and V n,α (1; x) = 1.In the special case α = 1, the operators V n,α reduce to the operators V n ≡ V n,1 .Some basic properties of J n,k are as follows: In this paper, we study the rate of convergence for the new sequence of operators (1.4), for functions f of bounded variation.Our result essentially generalizes and improves the results of [8,11].Furthermore, we find the limit of the sequence V n,α (f ; x) for bounded locally integrable functions f having a discontinuity of the first kind at x ∈ (0, ∞).

The main results.
As a main result, we derive the following estimate on the rate of convergence.Theorem 2.1.Assume that f ∈ W (0, ∞) is a function of bounded variation on every finite subinterval of (0, ∞).Furthermore, let α ≥ 1, λ > 2, and x ∈ (0, ∞) be given.Then, for each r ∈ N, there exists a constant M(f , α, r , x) such that for sufficiently large n, the Bézier-type Baskakov-Durrmeyer operators V n,α satisfy the estimate where Remark 2.2.The exponent r in the O-term of (2.1) can be chosen arbitrary large.
As an immediate consequence of Theorem 2.1, we obtain in the special case α = 1 the following estimate which improves the results of [8,11].
Corollary 2.3.Under the assumptions of Theorem 2.1, there holds, for sufficiently large n, where g x is defined as in Theorem 2.1. (2.4)

Auxiliary results.
In order to prove our main result, we will need the following lemmas.Throughout the paper, for each real x, let ψ x (t) = t − x.Lemma 3.1 (see [6]).Let {ξ i } ∞ i=1 be a sequence of independent and identically distributed random variables with finite variance such that the expectation Then there exists a constant c with 1/ √ 2π < c < 0.82 such that, for all n = 1, 2, 3,... and all t ∈ R, Lemma 3.2 (see [10]).For each fixed x ∈ [0, ∞) and m ∈ N 0 , the central moments In particular, Remark 3.3.Note that, given any λ > 2 and any x > 0, for all n sufficiently large, we have the estimate Lemma 3.4 (see [13]).For all x > 0 and n, k ∈ N, there holds Throughout, let With this definition, for each function f ∈ W (0, ∞), there holds, for all sufficiently large n, Note that, in particular, Lemma 3.5.For each λ > 2 and, for all sufficiently large n, there exist, for all x ∈ (0, ∞), Proof.First we prove (3.10).There holds where we applied Lemma 3.4.Now (3.10) is a consequence of Remark 3.3.The proof of (3.11) is similar.

Proofs of the main results
Proof of Theorem 2.1.Our starting point is the identity where δ x (t) = 1 (t = x) and δ x (t) = 0 (t ≠ x) (see [12,Equation (28)]).Since V n,α (δ x ; x) = 0, we conclude that First, we obtain we conclude that n,j (x) = 1.Therefore, we obtain By the mean value theorem, it follows that where J n−1,k+1 (x) < γ n,k (x) < J n−1,k (x).Hence, where In order to complete the proof of the theorem, we need an estimate of V n,α (g x ; x).We use the integral representation (3.8) and decompose [0, ∞) into three parts as follows: We start with and therefore Integrating the last term by parts, we get (4.17) Replacing the variable y in the last integral by x − x/ √ n, we obtain Hence, Finally, we estimate I 3 .We let and divide I 3 = I 31 + I 32 , where (4.21) With y = x + x/ √ n, the first integral can be written in the form (4.23) In a similar way as above we obtain 2x y which implies the estimate We proceed with I 32 .By assumption, there exists an integer r such that f (t) = O(t 2r ) as t → ∞.Thus, for a certain constant M > 0, depending only on f , x, and r , we have x given by ψ 2 x (t) = (t − x) 2 is of bounded variation on every finite subinterval of [0, ∞), we deduce from Theorem 2.1 that, for all x ∈ (0, ∞), 2) is also bounded and is continuous at the point x.By the Korovkin theorem, we conclude that Therefore, the right-hand side of inequality (4.2) tends to zero as n → ∞.This completes the proof of Theorem 2.4.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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(2. 2 )
and b a (g x ) is the total variation of g x on [a, b].