RATE OF CONVERGENCE ON BASKAKOV-BETA-BEZIER OPERATORS FOR BOUNDED VARIATION FUNCTIONS

by replacing the discrete value f(k/n) by the integral (n−1)∫∞ 0 pn,k(t)f (t)dt in order to approximate Lebesgue integrable functions on the interval [0,∞). Some approximation properties of the operators (1.1) were discussed in [6, 7, 8]. In [2, 3], the author defined another modification of the Baskakov operators with the weight functions of Beta operators so as to approximate Lebesgue integrable functions on [0,∞). For f ∈H[0,∞), Baskakov Beta operators are defined as

In [2,3], the author defined another modification of the Baskakov operators with the weight functions of Beta operators so as to approximate Lebesgue integrable functions on [0, ∞).For f ∈ H[0, ∞), Baskakov Beta operators are defined as (1.4) where p n,k (x) is as defined in (1.1), b n,k (t) = t k /(B(k + 1,n)(1 + t) n+k+1 ), and B(k It was observed in [2] that the integral modification of the Baskakov operators defined by (1.4) gives better results than the operators (1.1), and some approximation properties for the operators B n become simpler in comparison to the operators V n , for example, in the estimation of the rate of convergence of bounded variation functions we need not to use result of the type [8, Lemma 5] for the operators (1.4).This motivated further study of the operators B n .For f ∈ H[0, ∞), α ≥ 1, we introduce the Bezier variant of the operators (1.4) as follows: where is the Baskakov basis function.Obviously, B n,α (1,x) = 1, and, particularly when α = 1, the operators (1.4) reduce to the operators (1.4).It is observed that B n,α (f , x) is the sequence of linear positive operators.Some basic properties of J n,k (x) are as follows: (i) These properties can be obtained easily by direct computation.
Bojanić and Vulleumier [1] estimated the rate of convergence of Fourier series of functions of bounded variation.Recently, Zeng and Chen [11] estimated the rate of convergence of the Durrmeyer-Bezier operators for functions of bounded variation.Zeng and Gupta [12] and Zeng [10] estimated the rate of approximation for the Bezier variant of classical Baskakov and Szász operators, respectively, at those points at which one-sided limits f (x±) exist.In the present paper, we estimate the rate of convergence of the operators B n,α (f , x) for functions of bounded variation.The approximation properties for the operators B n,α (f , x) are different.), and let at a fixed point x ∈ (0, ∞), the one-sided limits f (x±), exist.Then, for α ≥ 1, λ > 2, x ∈ (0, ∞) and for n > max{1 + β, N(λ, x)}, we have where and V b a (g x ) is the total variation of g x on [a, b].

Auxiliary results.
In this section, we give certain results, which are necessary to prove the main result.
Yuankwei and Shunsheng [8] gave the following inequality for Baskakov basis functions.For x ∈ (0, ∞) and k ∈ N, there holds The above bound discussed in [8] was not sharp bound.Recently, Zeng [9] estimated the exact bounds for Bernstein basis functions and Meyer-Konig-Zeller basis functions.
Using the inequality estimate of Zeng [9], the exact bound for Baskakov basis functions can be obtained as in the following lemma.

Proof of Theorem 1.1.
It is easily verified [11] that In order to prove the theorem, we need the estimates for B n,α (g x ,x) and B n,α (sign(t − x), x).We first estimate B n,α (sign(t − x), x) as follows: Using Lemma 2.2, we have (3.3) Thus, By mean value theorem, we have where J α n,j (x) < γ α n,j (x) < J α n,j (x).Therefore, Using Lemma 2.1, we have Next, we estimate B n,α (g x ,x) as follows: and thus Next, we estimate E 1 .Setting y = x − x/ √ n and integrating by parts, we have Also, y = x − x/ √ n ≤ x, by Lemma 2.4, we get Integrating the last integral by parts, we obtain Replacing the variable y in the last integral by x − x/ √ n, we get (3.15) Hence, Finally, we estimate E 3 , and setting z = x + x/ √ n, we have (3.17) We define ∆ n,α (x, t) on [0, 2x] as Thus, Integrating by parts, we get where ∆n,α (x, t) is the normalized form of ∆ n,α (x, t).
Now, using Lemma 2.4 and the fact that ∆n,α (x, t) ≤ ∆ n,α (x, t) on [0, 2x], we have Thus, arguing similarly as in the estimate of E 1 , we get (3.23) Again, by Lemma 2.4, we have (3.24) Finally, for n > β, we have Using the identity we get This completes the proof of the theorem.
Remark 3.1.It is easier to define the Bezier variants of the well-known summationintegral type operators.For example, Szász-Mirakyan-Baskakov operators S n and Baskakov-Szász type operators M n were introduced and studied in [4,5], respectively.We may introduce their Bezier variants as follows.
But the analogous results for these operators are not possible.The main problem is in the estimation of S n,α (sign(t −x), x) and M n,α (sign(t −x), x) because we cannot relate summation of Szász (Baskakov) basis with integral of Baskakov (Szász) basis functions.That is, we cannot find result of the type [8, Lemma 5].There may be some other techniques to solve this problem.This problem is still unresolved, and it is an open problem for the readers.Remark 3.2.It was observed in [2] that the operators with weight functions of Beta operators give better results in simultaneous approximation than the usual Baskakov Durrmeyer operators studied in [7,8].Here, we have considered the weight functions of Beta operators, and, for α = 1, we obtain the better estimate on the rate of convergence for bounded variation functions over the main results of [8].

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.