) , 19 – 33 On a General Class of q-Rational Type Operators

In this study, we define a general class of rational type operators based on q-calculus and investigate the weighted approximation properties of these operators by using A-statistical convergence. We also estimate the rates of A-statistical convergence of these operators by modulus of continuity and Petree’s K-functional. The operators to be introduced, include some well known q-operators so our results are true in a large spectrum of these operators.

1. Introduction K. Balazs [3] introduced the Bernstein type rational functions and proved the convergence theorems for them.Later, K.Balazs and J.Szabados [4] improved some estimates on the order of approximation for the Bernstein type rational operators.
The generalization of the Bernstein type rational operators are introduced by C. Atakut and N. Ispir [16] as follows (1) L n (f where a n and b n are suitably chosen real numbers, independent of x.Here {ψ n } is a sequence of functions ψ n : C → C satisfying the following conditions: n (0) ≥ 0 for any n = 1, 2, . . ., k = 1, 2, . . .On a General Class of q-Rational Type Operators d) For every n = 1, 2, . . ., it is (2) lim where a n → 0, as n → ∞.In [16] the authors estimated the order of approximation for the operators defined by (1) and proved a Voronovskaja type asymptotic formula and pointwise convergence in simultaneous approximation.In [13,14,18] the approximation properties for different variants of the operators (1) were investigated in various function spaces .In [17], the approximation properties of the Kantorovich variant of the operators (1) were given by the aid of A-statistical convergence.Notice that A -statistical convergence is stronger than usual convergence.
It is known that the applications of q-calculus in the area of approximation theory have been an active area of research.In the recent years, the statistical approximation properties of some positive linear operators based on q-integers have been studied intensively by many authors (e.g.[5,15,17,21]).
The aim of this study is to introduce q-type generalization of the operators (1) and investigate the A-statistical approximation properties of the constructed operators in weighted spaces.Using A-statistical convergence, we obtain weighted Korovkin type theorem and weighted order of approximation by the constructed operators based q-calculus.Moreover, we estimate the rate of A-statistical convergence by usual modulus of continuity and by Petree's K-functional in the different normed spaces for q-extension of the operators (1).Now, let us give a few basic definitions and notations in q-integers shortly.Details on q-calculus can be found in [7].Throughout the present paper, we consider q as a real number such that 0 < q < 1, and for each nonnegative integer i, the q-integer [i] q is defined by For fixed 0 < q < 1, the q-derivative of a function f : R → R with respect to x is defined by The chain rule for ordinary derivatives is similar for q-derivative.
At this point, we recall the q-Taylor theorem in the following.Theorem A( [7], p. 103.)If a function f (x) possess convergence series expansion then where Now let us recall some concepts of the A-statistical convergence.Suppose that A is non-negative summability matrix and let K be subset of N the set of natural numbers.The A−density of K is defined by δ A (K) := lim j 1 n ∞ n=1 a jn χ K (n) provided limit exists, where χ K characteristic function of K.A sequence x = (x n ) is called A−statistically convergent to L if for every ε > 0 lim j n: |xn−L|≥ε a jn = 0 or equivalently for every ε > 0, δ A {k ∈ N: |x k − L| ≥ ε} = 0.In this case we write st A − lim x = L [8,9].
The case in which A = C 1 , the Cesáro matrix of order one, A -statistical convergence reduces to the statistical convergence [9,19].Also if A = I, the identity matrix, then it reduces to the ordinary convergence.We note that, if A = (a jn ) is a non-negative regular matrix such that lim j max n {a jn } = 0, then A-statistical convergence is stronger than convergence [19].It should be noted that the concept of A -statistical convergence may also be given in normed spaces: Assume (X, .) is a normed space and u = (u k ) is a X-valued sequence.Then (u k ) is said to be A-statistically convergent to u 0 ∈ X if, for every ε > 0, δ A {k ∈ N :

Construction of operators and auxiliary results
Now we would like to introduce q-generalization of the operators (1 ).Let (ϕ n ) be a sequence of real functions on R + which are continuously infinitely q−differentiable on R + satisfying the following conditions 1. ϕ n (0) = 1, for each n ∈ N 2. D k q ϕ n (0) ≥ 0, for every n, k ∈ N, x ≥ 0 3.For every n ∈ N, with For fixed x ∈ R + , taking account to Theorem A we get (4) , n ∈ N, we introduce qgeneralization of the operator (1) as follows (5) On a General Class of q-Rational Type Operators for each n ∈ N, x ≥ 0. Note that: • It is easily verified that the operators L n,q are linear positive operators.• The order of convergence is the best possible estimate for β ∈ (0, 2/3] (see [4]).• The present condition (3) is weaker than the present one given by (2) for q = 1.Indeed, we can construct a sequence such that it is statistically convergent to 1 but not convergent in the ordinary sense.
A well known example is defined as; , and α n = 1 otherwise.Same result also works to A−statistical convergence.
Proof.From (4) and definition of L n,q (f ) it is clear that L n,q (e 0 ) (x) = 1.
Considering (4 ), we can write the q-derivative of ϕ n with respect to x as where used the equation (a n x) k q = q k(k−1) (a n x) k .Hence multiplying both sides by x and dividing by x we obtain (10) which gives the (7).We use a similar technique to get (8).Again differentiating (9) with respect to x we have (11) [n] Using the equality [k − 1] q = [k] q − q k−1 and multiplying both sides by x 2 we have Dividing by x we write which gives the (8) by using formulas ( 5) and (10).

A-Statistical convergence in weighted spaces
Let ρ denotes a continuous weight function with ρ(x) ≥ 1, x ∈ [0, ∞) and ρ (x) → ∞ as x → ∞.Let B ρ be the weighted space of all functions f defined on the R + satisfying the condition |f (x)| ≤ M f ρ(x) with some constant M f , depending only on f .By C ρ , let us denote the subspace of all continuous functions belong to B ρ .Also, let C 0 ρ be the subspace of all functions f ∈ C ρ for which lim |x|→∞ f (x)/ρ(x) = 0. Endowed with the norm f ρ = sup x≥0 (|f (x)| /ρ (x)) these spaces are Banach spaces.Note that the weighted Korovkin type theorem were proved by A.D. Gadjiev [10,11].Using A-statistical convergence, the weighted Korovkin type theorem was given in [6].
Let {L n,q } be the sequence of linear positive operators defined by (5).Then it is easily seen that L n,q : C ρ → B ρ .
Let q = {q n } be a sequence satisfying the following conditions ( 12) The condition (12) guaranties that st A − lim n [n] −1 q = 0. Now we are ready to prove our first result which is related to the Astatistical convergence the sequence of {L n,qn (f )} to f .Theorem 3.1.Let A = (a jn ) be non-negative regular summability matrix, the sequence q = {q n } satisfies (12) with q n ∈ (0, 1] for all n ∈ N. Then for every Proof.From Lemma1, it is obvious that st A − lim n L n,q (e 0 ) − e 0 ρ = 0.
Using the (6), we get Now, for a given ε > 0, we define the sets U = n : L n,qn (e 1 ) − e 1 ρ ≥ ε and U 1 = {n : B n,qn (ϕ n , x) ≥ ε} .It is clear that U ⊂ U 1 and hence From the condition (3) we get st A − lim B n,qn (ϕ n , x) = 0. Therefore, it is clear that δ A {n ∈ N : B n,qn (ϕ n , x) ≥ ε} = 0 and hence we have Similarly from (8) we can write x and hence we get Now for given ε > 0, let us define the following sets From the condition (12) we get (13) st A − lim 1 Consequently we obtain that st A − lim L n,qn (e i ) − e i ρ = 0, i = 0, 1, 2 which completes the proof of the Theorem according to the weighted Korovkin type Theorem [12,6,10].
As a consequence, for all n ∈ N, x ≥ 0 and 0 < q n < 1, we have ( 14) L n,qn (e 1 − e 0 x) Theorem 3.2.Let A = (a jn ) be non-negative regular summability matrix, the sequence q = {q n } satisfies ( 12) with q n ∈ (0, 1] for all n ∈ N. If any function f ∈ C ρ , satisfies the Lipschitz condition that is where M is a constant.
Proof.Since L n,qn is a linear positive operator and f satisfies the Lipschitz condition we can write, Applying the Holder inequality with p = 2/α, s = 2/ (2 − α) and saying .
Taking account to (16) and using the conditions (3) and ( 12),similarly with the proof of the Theorem1, we obtain the desired result.Now, we concern with the order of approximation of a function f ∈ C 0 ρ by the linear positive operator L n,q .We will use the weighted modulus of continuity defined by The weighted modulus of continuity has the following properties (see [14]): Notice that, if f is not uniformly continuous on the interval [0, ∞); then the usual first modulus of continuity ω(f ; δ) does not tend to zero, as δ → 0. It is seen that Ω m (f ; δ) → 0,as δ → 0 for all f ∈ C 0 ρ due to the property (i).We now give second our main result.The following theorem is given an estimate for the approximation error with the operators L n,qn (f ), by means of Ω 1 (f ; δ) with ρ(x) = 1 + x.Theorem 3.3.Let {q n } be a sequence satisfying the condition (12) with q n ∈ (0, 1] for all n ∈ N. Suppose that the condition holds where ρ 2 (x) = 1 + x 2 and C is a constant independent of f and n.
Proof.Considering the definition of Ω 1 (f ; δ) and by using the property (iii) of Ω 1 (f ; δ) we can write On a General Class of q-Rational Type Operators Since L n,qn is a linear and positive operator we get ( 18) To estimate the first term, considering ( 6) and ( 7), we can write ( 19) Applying the Cauchy-Schwarz inequality to the second term in (18), since L n,qn is a linear and positive, we get (20) .
We now estimate the first term.By using ( 6), (7) and (8) and by simple calculations ,we get (21) L n,qn (1 + 2x + t) 2 (x) Taking into account ( 16), if we estimate the second term then we get with C 1 is a constant independent of n.Combining ( 19), ( 20), ( 21) and ( 22) with (18) we have We notice that, from (13), it is clear that st in the ρ 2 -norm.

Local Approximation
Theorem 4.1.Let {q n } be a sequence satisfying the condition (12) with q n ∈ (0, 1] for all n ∈ N. We have 1) For any f ∈ C ρ we have where ω (f, δ) is the usual first modulus of continuity of f and Proof. 1) Using the linearity and positivity of the operator L n,qn and the known properties of ω (f, δ) and applying Cauchy -Schwarz inequality we obtain ≤ ω(f ; δ) L n,qn (e 0 ) (x) + 1 δ L n,qn (e 1 − e 0 x) 2 (x) .
By choosing δ = δ n,x as in (23), we reach the desired result.Notice that, taking into account ( 14), we get st A − lim n δ n,x = 0 for all fixed x.

Concluding Remarks
Some particular cases of the operators L n,q are defined as follows: