On a New Type of q-Baskakov Operators

First, let us provide some background information on elements of q-analysis and on their well known formulas which are studied first by Euler in the eighteenth century. Following this, many interesting results in this field were obtained in the beginning of nineteenth century by F. H. Jackson [1] who introduced q-functions. He also developed q-calculus in a systematic way. Below, we give the concepts of q-integers, q-factorials, q-binomial coefficients, and q-derivative. The definitions used in this study are based on terminology and notations as seen in books [2], [3], and [4]. Let q > 0. The required q-Calculus theorems and definitions are as outlined below. The q-analogue of the integer m ∈ N, called q-integer, is given by


Introduction
First, let us provide some background information on elements of q-analysis and on their well known formulas which are studied first by Euler in the eighteenth century.Following this, many interesting results in this field were obtained in the beginning of nineteenth century by F. H. Jackson [1] who introduced q-functions.He also developed q-calculus in a systematic way.Below, we give the concepts of q-integers, q-factorials, q-binomial coefficients, and q-derivative.The definitions used in this study are based on terminology and notations as seen in books [2], [3], and [4].Let q > 0. The required q-Calculus theorems and definitions are as outlined below.The q-analogue of the integer m ∈ N, called q-integer, is given by Also [0] q := 0. Similarly, the q-analogue of the factorial of m is known by Now, let us give the q-version of the Gauss binomial formula.The analogue of (a + b) m in q-analysis, are given by By simple calculations, it follows that is the q-binomial formula.All the concepts defined above, become their classical cases if q tends to 1.
The derivative of a function f in q calculus, shown by D q f , is given by Let us define the partial derivatives of a function of two variables in q-calculus, say q-partial derivative: The q-partial derivative of f for the variable y can be given similarly.
In this study, we have introduced a new type of q-analogous of the classical Baskakov operators which are produced from general discrete type operators that are defined in the next section.Then, the approximation properties and the rate of convergence of the sequences of this operators have been established in terms of the modulus of smoothness of order 1.

Material and Method
Let q > 0 and m ∈ N.For f ∈ C(I) (I = [0, 1] or [0, ∞)), the authors introduced the following operators on C(I) in [19] and the approximation properties and the rate of convergence of these sequences of q-discrete type is established by means of the modulus of continuity in [20]: where α m,ν,q are positive numbers and ϕ m,q (x, u) generating real functions defined on I × [0, ∞), have the following conditions: (i) ϕ m,q (x, 0) = 0 and ϕ m,q (x, 1) = 1 for all m ∈ N and x ∈ I. (ii) exist and are continuous functions of x for all ν ∈ N 0 and m ∈ N.
(iii) For all ν ∈ N 0 , x, u ≥ 0, It is clear that the operators are positive and linear, thanks to (iii) on the space of bounded functions on I, B(I).
The test functions e r,i are given by e r,i (t The functions of e r,0 are used as test functions for q-Butzer-Bleimann-Hahn Operators, the functions e r,1 are used as test functions for q-Bernstein, q-Szasz-Mirakyan, q-Lupas and q-Baskakov Operators and the functions of e r,2 for q-Meyer-König and Zeller Operators.In continuation of the relation for the numbers α n,ν,q indicated in (10), we assume the following: where α m,q are positive numbers independent of ν.

Theorem 2.1 ([19]
).If the sequence ϕ m,q (x, u) satisfies the conditions (i)-(iii) for all r ∈ N 0 and m ∈ N, then the following relation is true: S q (r, ν) appeared in are the q-Stirling numbers of the second kind for detail, see [22].
Corollary 2.2.By virtue of equality (7), we have E m,q (e 0,i ; x) = 1; Now, let us construct the new type of q-analogue of Baskakov operators.For m ∈ N and q ∈ (0, 1), we consider the function ϕ m,q (x, 0) = 0 and ϕ m,q (x, 1) = 1 for every m ∈ N and x ∈ [0, ∞).By the definition of q-partial derivatives, we obtain E. Şimşek / On a New Type of q-Baskakov Operators So we have, By induction, we obtain that If we write u = 0 in the last equality, then we get Since the right hand of the equality ( 9) is a rational function of x which does not have any singular points in [0, ∞), then the condition (ii) holds and since q < 1 and x ∈ [0, ∞), then the condition (iii) is satisfied too, there by the functions ϕ m,q (x, u) defined by ( 8) generate some positive and linear operators.Writing ( 9) and considering α m,ν,q = [m] q in the operators E m,q given by ( 6), we have, for (10)

Results
In this section we give some classical approximation properties of the operators E m,q .Let q m ∈ (0, 1) and 1 − q m = o( 1 m ) when m → ∞.In the sequel for j ∈ N 0 , m ∈ N, we use notations: By simple calculations, we get the following lemmas.First, we get the following lemma from Corollary 2.2. ; And, we obtain the following results using Lemma 3.1.
for all t, x ∈ I A .
Proof.For the case r = 0 the assertion is obvious.We assume that r ∈ N.For t, x By monotonicity of the operators E m,q and using the Cauchy-Schwarz inequality, we have x for all m ∈ N 0 , thus we obtain E m,q (e r ; x) − e r (x) ≤ rA r−1 µ m,2 (x, q) what we wanted to prove.Proof.Indeed, Using following inequality Consequently, we have lim m→∞ µ m,2 (x, q m ) = 0.
From Lemma 3.3 we obtain Lemma 3.4.
Corollary 3.5.If f ∈ C(I A ), then we have For f ∈ C(I A ) and δ > 0, the modulus of smoothness of order 1 (it is also called modulus of continuity) of f with step δ > 0 given by For any δ > 0, we get Theorem 3.6.Let E m,q be given by ( 10) and f ∈ C(I A ), then the inequality holds for any δ > 0. Proof.Since E m,q (1; x) = 1, then we have for all n ∈ N. Now using (11) in inequality (12) we obtain for all δ > 0. Using the Cauchy-Schwartz Inequality and (13) it follows that As always, we write f ∈ Lip M α, (M > 0, 0 < α ≤ 1), if the relation is satisfied for all u, v ∈ I A .

Discussion and Conclusion
The type of q-Baskakov Operators which we constructed above is new and different from ones exists in literature.
The results which are obtained enrich the literate of convergence in q-calculus in operator theory.Consequently, the operators so established can be found fruitful in several situation appearing in the literature on Approximation Theory and Operator Theory.