On the convergence of a kind of q-gamma operators

In recent years, the applications of q-calculus in the approximation theory is one of the main areas of research. After q-Bernstein polynomials were introduced by Phillips [1] in 1997, many researchers have performed studies in this field; we mention some of them [1‐4].

In 2007, Karsli [5] introduced and estimated the rate of convergence for functions with derivatives of bounded variation on [ 0 , ∞ ) of new gamma type operators as follows:

In 2009, Karsli, Gupta and Izgi [6] gave an estimate of the rate of pointwise convergence of these operators (1) on a Lebesgue point of bounded variation function f defined on the interval ( 0 , ∞ ). In 2010, Karsli and Özarslan [7] obtained some direct local and global approximation results and gave a Voronoskaya-type theorem for the operators (1). As the application of q-calculus in approximation theory is an active field, it seems there are no papers mentioning the q analogue of these operators defined in (1). Inspired by Aral and Gupta [2], they defined a generalization of q-Baskakov type operators using q-Beta integral and obtained some important approximation properties, which motivates us to introduce the q analogue of this kind of gamma operators.

Before introducing the operators, we mention certain definitions based on q-integers; details can be found in [8, 9]. For any fixed real number 0 < q ≤ 1 and each nonnegative integer k, we denote q-integers by [ k ] q, where

Also, q-factorial and q-binomial coefficients are defined as follows:

The q-improper integrals are defined as (see [10])

provided the sums converge absolutely.

The q-beta integral is defined by

where K ( x ; t ) = 1 x + 1 x t ( 1 + 1 x ) q t ( 1 + x ) q 1 − t and ( a + b ) q τ = ∏ j = 0 τ − 1 ( a + q j b ), τ > 0.

In particular for any positive integer m, n,

where Γ q ( t ) is the q-gamma function satisfying the following functional equations:

(see [3]).

For f ∈ C [ 0 , ∞ ), q ∈ ( 0 , 1 ) and n ∈ N, we introduce a kind of q-gamma operators G n , q ( f ; x ) as follows:

Note that for q → 1 −, G n , 1 − ( f ; x ) become the gamma operators defined in (1).

In order to obtain the approximation properties of the operators G n , q, we need the following lemmas.

Lemma 1 For any k ∈ N, k ≤ n + 2 and q ∈ ( 0 , 1 ), we have

Proof Using the properties of q-beta integral, we have

Lemma 1 is proved. □

Lemma 2 The following equalities hold:

Proof From Lemma 1, taking k = 0 , 1 , 2 , 3 , 4, we get (6) and (7). Since G n , q ( ( t − x ) 2 ; x ) = G n , q ( t 2 ; x ) − 2 x G n , q ( t ; x ) + x 2 and G n , q ( ( t − x ) 4 ; x ) = G n , q ( t 4 ; x ) − 4 x G n , q ( t 3 ; x ) + 6 x 2 G n , q ( t 2 ; x ) − 4 x 3 G n , q ( t ; x ) + x 4, using (6), (7), we obtain (8) and (9) easily. □

Remark 1 Note that for q → 1 −, from Lemma 2, we have

which is the moments and central moments of the operators defined in (1).

In this section we establish direct and local approximation theorems in connection with the operators G n , q ( f , x ).

We denote the space of all real-valued continuous bounded functions f defined on the interval [ 0 , ∞ ) by C B [ 0 , ∞ ). The norm ∥ ⋅ ∥ on the space C B [ 0 , ∞ ) is given by ∥ f ∥ = sup { | f ( x ) | : x ∈ [ 0 , ∞ ) }.

Further, let us consider Peetre’s K-functional

where δ > 0 and W 2 = { g ∈ C B [ 0 , ∞ ) : g ′ , g ″ ∈ C B [ 0 , ∞ ) }.

For f ∈ C B [ 0 , ∞ ), the modulus of continuity of second order is defined by

By [[11], p.177] there exists an absolute constant C > 0 such that

Our first result is a direct local approximation theorem for the operators G n , q ( f , x ).

Theorem 1 For q ∈ ( 0 , 1 ), x ∈ [ 0 , ∞ ) and f ∈ C B [ 0 , ∞ ), we have

where C is a positive constant,

Let us define the auxiliary operators

. The operators G ˜ n , q ( f ; x ) are linear and preserve the linear functions

(see (6)).

Let g ∈ C B 2. By Taylor’s expansion

and (14), we get

Hence, by (13) and (8), we have

On the other hand, by (13), (4) and (6), we have

Now (13) and (15) imply

Hence taking infimum on the right-hand side over all g ∈ W 2, we get

By (10), for every q ∈ ( 0 , 1 ), we have

where α n , q ( x ) and β n , q ( x ) are defined in (12). This completes the proof of Theorem 1. □

Remark 2 Let q = { q n } be a sequence satisfying 0 < q n < 1 and lim n q n = 1, we have lim n α n , q n = 0 and lim n β n , q n ( x ) = 0, these give us the pointwise rate of convergence of the operators G n , q n ( f ; x ) to f ( x ).

Let B x 2 [ 0 , ∞ ) be the set of all functions f defined on [ 0 , ∞ ) satisfying the condition | f ( x ) | ≤ M f ( 1 + x 2 ), where M f is a constant depending only on f. We denote the subspace of all continuous functions belonging to B x 2 [ 0 , ∞ ) by C x 2 [ 0 , ∞ ). Also, let C x 2 ∗ [ 0 , ∞ ) be the subspace of all functions f ∈ C x 2 [ 0 , ∞ ), for which lim x → ∞ f ( x ) 1 + x 2 is finite. The norm on C x 2 ∗ [ 0 , ∞ ) is ∥ f ∥ x 2 = sup x ∈ [ 0 , ∞ ) | f ( x ) | 1 + x 2. We denote the usual modulus of continuity of f on the closed interval [ 0 , a ] (a > 0) by

Obviously, for function f ∈ C x 2 [ 0 , ∞ ), the modulus of continuity ω a ( f , δ ) tends to zero.

Theorem 2 Let f ∈ C x 2 [ 0 , ∞ ), q ∈ ( 0 , 1 ) and ω a + 1 ( f , δ ) be the modulus of continuity on the finite interval [ 0 , a + 1 ] ⊂ [ 0 , ∞ ), where a > 0. Then we have

Proof For x ∈ [ 0 , a ] and t > a + 1, we have

hence we obtain

For x ∈ [ 0 , a ] and t ≤ a + 1, we have

From (17) and (18), we get

For x ∈ [ 0 , a ] and t ≥ 0, by Schwarz’s inequality and Lemma 2, we have

By taking δ = 2 a 1 − q [ n + 1 ] q [ n + 2 ] q, we get the assertion of Theorem 2. □

Now we will discuss the weighted approximation theorem.

Theorem 3 Let the sequence q = { q n } satisfy 0 < q n < 1 and q n → 1 as n → ∞, for f ∈ C x 2 ∗ [ 0 , ∞ ), we have

Proof By using the Korovkin theorem in [12], we see that it is sufficient to verify the following three conditions.

Since G n , q n ( 1 ; x ) = 1 and G n , q n ( t 2 ; x ) = x 2, (20) holds true for v = 0 and v = 2.

Finally, for v = 1, we have

since lim n → ∞ q n = 1, we get lim n → ∞ q [ n + 1 ] q [ n + 2 ] q = 1, so the condition of (21) holds for v = 1 as n → ∞, then the proof of Theorem 3 is completed. □

Finally, we give a Voronovskaya-type asymptotic formula for G n , q ( f ; x ) by means of the second and fourth central moments.

Theorem 4 Let q : = { q n } be a sequence satisfying 0 < q n < 1, lim n q n = 1 and lim n q n n = 1. For f ∈ C x 2 2 [ 0 , ∞ ), (f ( x ) is a twice differentiable function in [ 0 , ∞ )), the following equality holds

for every x ∈ [ 0 , A ], A > 0.

Proof Let x ∈ [ 0 , ∞ ) be fixed. By the Taylor formula, we may write

where r ( t ; x ) is the Peano form of the remainder, r ( t ; x ) ∈ C x 2 [ 0 , ∞ ) and using L’Hopital’s rule, we have

Applying G n , q ( f ; x ) to (23), we obtain

By the Cauchy-Schwarz inequality, we have

Since r 2 ( x ; x ) = 0, then it follows from Theorem 3 that

Now, from (24), (25) and Lemma 2, we get immediately

and since q n + 1 = [ n + 2 ] q − [ n + 1 ] q ≤ [ n + 2 ] q − q [ n + 1 ] q ≤ [ n + 2 ] q − q [ n + 1 ] q = 1, we have

Theorem 4 is proved. □