R E S E A R C H
Open Access
Chlodowsky variant of
q-Bernstein-Schurer-Stancu operators
Tuba Vedi
*and Mehmet Ali Özarslan
*Correspondence:
tuba.vedi@emu.edu.tr
Eastern Mediterranean University, Gazimagusa, Mersin, TRNC 10, Turkey
Abstract
It was Chlodowsky who considered non-trivial Bernstein operators, which help to approximate bounded continuous functions on the unbounded domain. In this paper, we introduce the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. By obtaining the first few moments of these operators, we prove Korovkin-type approximation theorems in different function spaces. Furthermore, we compute the error of the approximation by using the modulus of continuity and Lipschitz-type functionals. Then we obtain the degree of the approximation in terms of the modulus of continuity of the derivative of the function. Finally, we study the generalization of the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators and investigate their approximations.
MSC: Primary 41A10; 41A25; secondary 41A36
Keywords: Schurer-Stancu and Schurer Chlodowsky operators; modulus of
continuity; Korovkin-type theorems; Lipschitz-type functionals; q-Bernstein operators
1 Introduction
The classical Bernstein-Chlodowsky operators were defined by Chlodowsky [] as
Cn(f ; x) = n r= f r nbn n r x bn r – x bn n–r ,
where the function f is defined on [,∞) and (bn) is a positive increasing sequence with bn→ ∞ and bnn → as n → ∞.
In , Stancu [] constructed and studied the Bernstein-Stancu operators, which were defined as Pαm,β(f ; x) = m k= f k+ α m+ β m k xk( – x)m–k,
where α, β∈ R such that ≤ α ≤ β.
In , Lupaş [] introduced the q-based Bernstein operators and obtained the Korovkin-type approximation theorem. In , other q-based Bernstein operators were defined by Phillips [, ]. During the last decade q-based operators have become an active research area (see [–]).
©2014Vedi and Ali Özarslan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In , the q-Bernstein-Schurer operators were defined by Muraru [], for fixed p∈ N
and for all x∈ [, ], by
Bpn(f ; q; x) = n+p k= f [k] [n] n+ p k xk n+p–k– s= – qsx , (.)
where q is a positive real number and the function f is evaluated at the q-integers [k][n]. Recall that the q-integer of k∈ R is []
[k] = ⎧ ⎨ ⎩ ( – qk)/( – q), q= , k, q= ,
the q-factorial is defined by
[k]! = ⎧ ⎨ ⎩ [k][k – ]· · · [], k = , , , . . . , , k= and q-binomial coefficients are defined by
n k = [n]! [n – k]![k]!
for n≥ , k ≥ . Note that the case q → –in (.), the operators defined by (.) were reduced to the operators considered by Schurer [].
Some properties of the q-Bernstein-Schurer operators were investigated in []. We should notice that the case p = reduces to the q-Bernstein operators.
It should be noted that complex approximation properties of some Schurer-type oper-ators were investigated in [, ] and [].
Recently, Barbosu investigated Schurer-Stancu operators Sαm,β,p : C[, + p]→ C[, ]
which were defined by [] (see also [])
Sαm,β,p(f ; x) = m+p k= f k+ α m+ β m+ p k xk( – x)m+p–k, (.)
where α and β are non-negative numbers which satisfy ≤ α ≤ β and also p is a non-negative integer.
In , Karslı and Gupta [] defined the q-analogue of Chlodowsky operators by
Cn(f ; q; x) = n k= f [k] [n]bn n k x bn k n–k– s= – qs x bn , ≤ x ≤ bn,
where (bn) has the same property of Bernstein-Chlodowsky operators.
Lately, the q-analogues of Bernstein-Schurer-Stancu operators were introduced by Agrawal et al. as [] Sαn,p,β(f ; q; x) = n+p k= f [k] + α [n] + β n+ p k xk n+p–k– s= – qsx ,
where α, β and p are non-negative integers such that ≤ α ≤ β. For the first few moments, we have the following lemma.
Lemma .[, p.] For the operator Sαn,p,β(f ; q; x), we have the following moments:
(i) Sα,β n,p(; q; x) = , (ii) Sαn,p,β(t; q; x) =[n + p]x + α [n] + β , (iii) Sαn,p,βt; q; x = ([n] + β) [n + p]x+ [n + p]x( – x) + α[n + p]x + α . The organization of the paper as follows.
In Section , we introduce the Chlodowsky variant of q-Bernstein-Schurer-Stancu oper-ators and investigate the moments of the operator. In Section , we study several Korovkin-type theorems in different function spaces. In Section , we obtain the order of conver-gence of the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators by means of Lipschitz class functions and the first modulus of continuity. In addition, we calculate the degree of convergence of the approximation process in terms of the first modulus of continuity of the derivative of the function. In Section , we study the generalization of the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators and investigate their approximations.
2 Construction of the operators
We construct the Chlodowsky variant of q-Bernstein-Schurer-Stancu operators as
Cn(α,β),p (f ; q; x) := n+p k= f [k] + α [n] + βbn n+ p k x bn k n+p–k– s= – qs x bn , (.)
where n∈ N and p ∈ Nand α, β∈ R with ≤ α ≤ β, ≤ x ≤ bn, < q < . Clearly, C(α,β)n,p
is a linear and positive operator. Note that the cases q→ and p = in (.) reduce to the Stancu-Chlodowsky polynomials [].
Firstly, we obtain the following lemma, which will be used throughout the paper.
Lemma . Let C(α,β)n,p (f ; q; x) be given in (.). Then the first few moments of the operators
are (i) Cn(α,β),p (; q; x) = , (ii) Cn(α,β),p (t; q; x) =[n + p]x + αbn [n] + β , (iii) Cn(α,β),p t; q; x = ([n] + β) [n + p – ][n + p]qx+ (α + )[n + p]bnx+ αbn, (iv) Cn(α,β),p (t – x); q; x = [n + p] [n] + β – x+ αbn [n] + β, (v) Cn(α,β),p (t – x); q; x = [n + p – ][n + p]q ([n] + β) – [n + p] [n] + β+ x + (α + )[n + p] ([n] + β) – α [n] + β bnx+ αb n ([n] + β).
Proof Using Lemma . and considering the following facts: Cnα,p,β(; q; x) = Sαn,p,β ; q; x bn , Cnα,p,β(t; q; x) = bnSαn,p,β t; q; x bn , Cnα,p,βt; q; x = bnSαn,p,β t; q; x bn , we get the assertions (i), (ii), and (iii).
Using the linearity of the operators, we have
Cn(α,β),p (t – x); q; x = Cn(α,β),p (t; q; x) – xCn(α,β),p (; q; x) = [n + p] [n] + β – x+ αbn [n] + β. So, we get (iv).
Similar computations give
Cn(α,β),p (t – x); q; x = Cn(α,β),p t; q; x – xCn(α,β),p (t; q; x) + xCn(α,β),p (; q; x). Then we have Cn(α,β),p (t – x); q; x = [n + p – ][n + p] ([n] + β) q– [n + p] [n] + β + x + (α + )[n + p] ([n] + β) – α [n] + β bnx+ αb n ([n] + β). This proves (v).
Lemma . For each fixed q∈ (, ), we have
[n + p – ][n + p] ([n] + β) q– [n + p] [n] + β + ≤ (qn[p] – β) ([n] + β) .
Proof Since [n + p – ][n + p]q≤ [n + p]we get
[n + p – ][n + p] ([n] + β) q– [n + p] [n] + β + ≤ [n + p] [n] + β – .
If we calculate the right-hand side of the above inequality, we get the following: [n + p – ][n + p] ([n] + β) q– [n + p] [n] + β + ≤ ([n] + β) ( – qn+p) ( – q) – ( – qn+p)( – qn) ( – q) – β ( – qn+p) – q + ( – qn) ( – q) + β( – q n) – q + β
= ([n] + β) qn(qp– qp+ ) ( – q) + β qn(qp– ) – q + β = ([n] + β) qn( – qp) ( – q) – β qn( – qp) – q + β = ([n] + β) qn[p]– βqn[p] + β=(q n[p] – β) ([n] + β) .
Remark . As a consequence of Lemma . and Lemma ., we have
Cn(α,β),p (t – x); q; x ≤(q n[p] – β) ([n] + β) x + (α + )[n + p] ([n] + β) bnx+ α b n ([n] + β). (.)
Lemma . For the second central moment we have the following estimate: sup ≤x≤bn Cn(α,β),p (t – x); q; x ≤(qn[p] – β) ([n] + β) b n+ (α + )[n + p] ([n] + β) bn+ α b n ([n] + β).
Proof Taking supremum over [, bn] in (.) we get the result.
3 Korovkin-type approximation theorem
In this section, we prove Korovkin-type approximation theorem for the Chlodowsky vari-ant of q-Bernstein-Schurer-Stancu operators. Denote by Cρ the space of all continuous
functions f , satisfying the condition f(x) ≤Mfρ(x), –∞ < x < ∞.
Obviously, Cρis a linear normed space with the norm
f ρ= sup
–∞<x<∞
|f (x)|
ρ(x).
In studying the weighted approximation, the following theorem is crucial.
Theorem .(See []) There exists a sequence of positive linear operators Tn, acting from Cρto Cρ, satisfying the conditions
lim n→∞Tn(; x) – ρ= , (.) lim n→∞Tn(φ; x) – φρ= , (.) lim n→∞Tn φ; x – φρ= , (.)
where φ(x) is continuous and increasing function on (–∞, ∞) such that limx→±∞φ(x) =
±∞ and ρ(x) = +φand there exists a function f∗∈ C
ρfor which limn→∞Tnf∗–f∗ρ> .
The following theorem has been given in [] and will be used in the investigation of the approximation properties of Cn(α,β),p (f ; q; x) in weighted spaces.
Theorem .(See []) The conditions (.), (.), (.) imply limn→∞Tnf– fρ= for any function f belonging to the subset C
ρof Cρfor which lim |x|→∞ f(x) ρ(x) exists finitely.
Particularly, choosing ρ(x) = + xand applying Theorem . to the operators
Tnα,β(f ; q; x) = ⎧ ⎨ ⎩ Cn(α,β),p (f ; q; x) if ≤ x ≤ bn, f(x) if x /∈ [, bn],
we can state the following theorem.
Theorem . For all f∈ C
+xwe have lim n→∞≤x≤bsupn |Tα,β n (f ; qn; x) – f (x)| + x =
provided that q:= (qn) with < qn< , limn→∞qn= and limn→∞[n]bn = as n→ ∞. Proof In the proof we directly use Theorem .. Obviously, by Lemma .(i), (ii), and (iii) we get the following inequalities, respectively:
sup ≤x≤bn |Tα,β n (t; qn; x) – x| + x ≤ sup≤x≤b n |([n+p] [n]+β – )|x + αbn [n]+β ( + x) ≤[n + p] [n] + β – + αbn [n] + β → and sup ≤x≤bn |Tα,β n (t; qn; x) – x| + x ≤ sup ≤x≤bn |[n + p – ][n + p]qn– ([n] + β)|x+ (α + )[n + p]bnx+ αbn ([n] + β)( + x) ≤|[n + p – ][n + p]qn– ([n] + β)| + (α + )[n + p]bn+ αbn ([n] + β) →
is satisfied since limn→∞qn= and [n]bn → as n → ∞.
Lemma . Let A be a positive real number independent of n and f be a continuous func-tion which vanishes on[A,∞]. Assume that q := (qn) with < qn< , limn→∞qnn= K <∞ and limn→∞b n [n]= . Then we have lim n→∞≤x≤bsupnC (α,β) n,p (f ; qn; x) – f (x)= .
Proof From the hypothesis on f , one can write|f (x)| ≤ M (M > ). For arbitrary small ε> , we have f[n] + β[k] + αbn – f (x) < ε +M δ [k] + α [n] + βbn– x ,
where x∈ [, bn] and δ = δ(ε) are independent of n. With the help of the following equality: n+p k= [k] + α [n] + βbn– x n+ p k x bn k n+p–k– s= – qsx bn = Cn(α,β),p (t – x); qn; x ,
we get by Theorem . and Remark . sup ≤x≤bn C(α,β)n,p (f ; qn; x) – f (x) ≤ ε +M δ (qn n[p] – β) ([n] + β) b n+ (α + )[n + p] ([n] + β) bn+ α b n ([n] + β) . Since bn
[n] → as n → ∞, we have the desired result.
Theorem . Let f be a continuous function on the semi-axis[,∞) and lim
x→∞f(x) = kf<∞.
Assume that q:= (qn) with < qn< , limn→∞qn= , limn→∞qnn= K <∞, and limn→∞b
n [n]= . Then lim x→∞≤x≤bsupnC (α,β) n,p (f ; qn; x) – f (x)= .
Proof The proof will be given along the lines of the proof of Theorem . in []. Clearly, it is sufficient to prove the theorem for the case kf = . Since limx→∞f(x) = , given any ε> we can find a point xsuch that
f(x) ≤ε, x≥ x. (.)
For any fixed c > , define an auxiliary function as follows:
g(x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ f(x), ≤ x ≤ x, f(x) –f(xc)(x – x), x≤ x ≤ x+ c, , x≥ x+ c.
Then for sufficiently large n in such a way that bn≥ x+c and in view of supx≤x≤x+c|g(x)| = |f (x)|, we have sup ≤x≤bn f(x) – g(x) ≤ sup x≤x≤x+c f(x) – g(x)+ sup bn≥x≥x+c f(x) ≤ sup x≤x≤x+c f(x)+ sup bn≥x≥x+c f(x).
We have from (.) sup
≤x≤bn
f(x) – g(x) ≤ε. Now, we can write
sup ≤x≤bn C(α,β)n,p (f ; qn; x) – f (x) ≤ sup ≤x≤bn Cn(α,β),p |f – g|; qn; x + sup ≤x≤bn Cn(α,β),p (g; qn; x) – g(x) + sup ≤x≤bn f(x) – g(x) ≤ ε + sup ≤x≤bn C(α,β)n,p (g; qn; x) – g(x),
where g(x) = for x+ c≤ x ≤ bn. By Lemma ., we obtain the result.
4 Order of convergence
In this section, we compute the rate of convergence of the operators in terms of the el-ements of Lipschitz classes and the modulus of continuity of the function. Additionally, we calculate the order of convergence in terms of the first modulus of continuity of the derivative of the function.
Now, we give the rate of convergence of the operators Cn(α,β),p in terms of the Lipschitz
class LipM(γ ), for < γ ≤ . Let CB[,∞) denote the space of bounded continuous
func-tions on [,∞). A function f ∈ CB[,∞) belongs to LipM(γ ) if
f(t) – f (x) ≤M|t – x|γ t, x∈ [, ∞) is satisfied.
Theorem . Let f ∈ LipM(γ ) C(α,β)n,p (f ; q; x) – f (x) ≤Mλn(x) γ/ , where λn,q(x) = ((q n n[p]–β) ([n]+β) )x+ ( (α+)[n+p] ([n]+β) –[n]+βα )bnx+ αbn ([n]+β).
Proof Considering the monotonicity and the linearity of the operators, and taking into account that f ∈ LipM(γ ),
C(α,β)n,p (f ; q; x) – f (x) = n+p k= f [k] + α [n] + βbn – f (x) n+ p k x bn k n+p–k– s= – qs x bn ≤ n+p k= f[k] + α[n] + βbn – f (x) n+ p k x bn k n+p–k– s= – qsx bn ≤ M n+p k= [n] + β[k] + αbn– x γ n+ p k x bn k n+p–k– s= – qsx bn .
Using Hölder’s inequality with p =γ and q =–γ , we get by the statement (.) C(α,β) n,p (f ; q; x) – f (x) ≤ M n+p k= [k] + α [n] + βbn– x n+ p k x bn k n+p–k– s= – qsx bn γ × n+ p k x bn k n+p–k– s= – qsx bn –γ ≤ M n+p k= [k] + α [n] + βbn– x n+ p k x bn k n+p–k– s= – qsx bn γ × n+p k= n+ p k x bn k n+p–k– s= – qs x bn –γ γ = MCn(α,β),p (t – x); q; x γ ≤ Mλn,q(x) γ .
Now we give the rate of convergence of the operators by means of the modulus of con-tinuity which is denoted by ω(f ; δ). Let f ∈ CB[,∞) and x ≥ . Then the definition of the
modulus of continuity of f is given by
ω(f ; δ) = max
|t–x|≤δ
t,x∈[,∞)
f(t) – f (x). (.) It follows that for any δ > the inequality
f(x) – f (y) ≤ω(f ; δ) |x – y| δ + (.) is satisfied []. Theorem . If f ∈ CB[,∞), we have C(α,β)n,p (f ; q; x) – f (x) ≤ωf; λn,q(x) ,
where ω(f ;·) is modulus of continuity of f which is defined in (.) and λn,q(x) be the same
as in Theorem..
Proof By the triangular inequality, we get C(α,β) n,p (f ; q; x) – f (x) = n+p k= f [k] + α [n] + βbn n+ p k x bn k n+p–k– s= – qsx bn – f (x) ≤ n+p k= f[k] + α[n] + βbn – f (x) n+ p k x bn k n+p–k– s= – qsx bn .
Now using (.) and the Hölder inequality, we can write C(α,β)n,p (f ; q; x) – f (x) ≤ n+p k= |[k]+α [n]+βbn– x| λ + ω(f ; λ) n+ p k x bn k n+p–k– s= – qs x bn = ω(f ; λ) n+p k= n+ p k x bn k n+p–k– s= – qsx bn +ω(f ; λ) λ n+p k= [n] + β[k] + αbn– x n+ p k x bn k n+p–k– s= – qs x bn = ω(f ; λ) +ω(f ; λ) λ n+p k= [k] + α [n] + βbn– x n+ p k x bn k n+p–k– s= – qs x bn / = ω(f ; λ) +ω(f ; λ) λ Cn(α,β),p (t – x) ; q; x/.
Now choosing λn,q(x) the same as in Theorem ., we have
C(α,β)n,p (f ; q; x) – f (x) ≤ωf;
λn,q(x)
.
Now, we compute the rate of convergence of the operators Cn(α,β),p in terms of the modulus
of continuity of the derivative of the function.
Theorem . If f(x) has a continuous bounded derivative f(x) and ω(f; δ) is the modulus
of continuity of f(x) in [, A], then f(x) – C(α,β)n,p (f ; q; x) ≤ M[n + p] [n] + β – A +[n] + βαbn + Bn,q(α, β) / ωf;Bn,q(α, β) / ,
where M is a positive constant such that|f(x)| ≤ M.
Proof Using the mean value theorem, we have
f [k] + α [n] + βbn – f (x) = [k] + α [n] + βbn– x f(ξ ) = [k] + α [n] + βbn– x f(x) + [k] + α [n] + βbn– x f(ξ ) – f(x) , where ξ is a point between x and [k]+α[n]+βbn. By using the above identity, we get
Cn(α,β),p (f ; q; x) – f (x) = f(x) n+p k= [k] + α [n] + βbn– x n+ p k x bn k n+p–k– s= – qs x bn + n+p k= [k] + α [n] + βbn– x f(ξ ) – f(x) n+ p k x bn k n+p–k– s= – qsx bn .
Hence, C(α,β)n,p (f ; q; x) – f (x) ≤f(x)Cn(α,β),p (t – x); q; x + n+p k= [n] + β[k] + αbn– xf(ξ ) – f(x) n+ p k x bn k n+p–k– s= – qs x bn ≤ M[n + p] [n] + β – A +[n] + βαbn + n+p k= [n] + β[k] + α– xf(ξ ) – f(x) n+ p k x bn k n+p–k– s= – qs x bn ≤ M[n + p] [n] + β – A +[n] + βαbn + n+p k= ωf; δ | [k]+α [n]+βbn– x| δ + [n] + β[k] + αbn– x n+ p k x bn k × n+p–k– s= – qsx bn , since |ξ – x| ≤[k] + α [n] + βbn– x . Using the above inequality, we have
C(α,β)n,p (f ; q; x) – f (x) ≤ M[n + p] [n] + β – A + αbn [n] + β + ωf; δ n+p k= [n] + β[k] + αbn– x n+ p k x bn k n+p–k– s= – qs x bn +ω(f ; δ) δ n+p k= [k] + α [n] + βbn– x n+ p k x bn k n+p–k– s= – qsx bn .
Therefore, applying the Cauchy-Schwarz inequality for the second term, we get C(α,β) n,p (f ; q; x) – f (x) ≤ M[n + p] [n] + β – A + αbn [n] + β + ωf; δ n+p k= [k] + α [n] + βbn– x n+ p k x bn k n+p–k– s= – qsx bn / +ω(f ; δ) δ n+p k= [k] + α [n] + βbn– x n+ p k x bn k n+p–k– s= – qsx bn
= M[n + p] [n] + β – A +[n] + βαbn + ωf; δ Cn(α,β),p (t – x); q; x +ω(f; δ) δ C (α,β) n,p (t – x); q; x . Therefore, using (.) we see that
sup ≤x≤AC (α,β) n,p (t – x); q; x ≤ sup ≤x≤A (qn[p] – β) ([n] + β) x + (α + )[n + p] ([n] + β) – α [n] + β bnx+ α b n ([n] + β) ≤(qn[p] – β) ([n] + β) A +(α + )[n + p] ([n] + β) – α [n] + β Abn+ αb n ([n] + β) := Bn,q(α, β). Thus, C(α,β)n,p (f ; q; x) – f (x) ≤M[n + p] [n] + β– A +[n] + βαbn + ωf; δ Bn,q(α, β) / + δBn,q(α, β) .
Choosing δ := δn,q(p) = (Bn,q(α, β))/, we obtain the desired result. 5 Generalization of the Chlodowsky variant of q-Bernstein-Schurer-Stancu
operators
In this section, we introduce generalization of Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. The generalized operators help us to approximate the con-tinuous functions on more general weighted spaces. Note that this kind of generalization was considered earlier for the Bernstein-Chlodowsky polynomials [] and q-Bernstein-Chlodowsky polynomials [].
For x≥ , consider any continuous function ω(x) ≥ and define
Gf(t) = f (t) + t
ω(t).
Let us take into account the generalization of the C(α,β)n,p (f ; q; x) as follows:
Lαn,p,β(f ; q; x) = ω(x) + x n+p k= Gf [k] + α [n] + βbn n+ p k x bn k n+p–k– s= – qsx bn ,
where ≤ x ≤ bnand (bn) has the same properties as the Chlodowsky variant of the
q-Bernstein-Schurer-Stancu operators.
Theorem . For the continuous functions satisfying
lim
x→∞
f(x)
we have lim n→∞≤x≤bsupn |Lα,β n,p(f ; q; x) – f (x)| ω(x) = . Proof Clearly, Lαn,p,β(f ; q; x) – f (x) = ω(x) + x n+p k= Gf [k] + α [n] + βbn n+ p k x bn k n+p–k– s= – qsx bn – Gf(x) , thus sup ≤x≤bn |Lα,β n,p(f ; q; x) – f (x)| ω(x) = sup≤x≤bn |C(α,β) n,p (Gf; q; x) – Gf(x)| + x .
By using|f (x)| ≤ Mfω(x) and continuity of the function f , we get|Gf(x)| ≤ Mf( + x) for x≥ and Gf(x) is continuous function on [,∞). Thus, from Theorem . we get the
result.
Finally note that, taking ω(x) = + x, the operators Lα,β
n,p(f ; q; x) reduce to Cn(α,β),p (Gf; q; x).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript. Received: 14 October 2013 Accepted: 20 January 2014 Published:13 May 2014
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10.1186/1029-242X-2014-189
Cite this article as: Vedi and Ali Özarslan: Chlodowsky variant of q-Bernstein-Schurer-Stancu operators. Journal of
