On approximation by Stancu type q-Bernstein-Schurer-Kantorovich operators

0.17 MB·

by  M. Mursaleen

On approximation by Stancu type
q-Bernstein-Schurer-Kantorovich operators
M. Mursaleen and Taqseer Khan
Department of Mathematics, Aligarh Muslim University, Aligarh–202002, India
[email protected]; [email protected];
6
1
0 Abstract
2

n
InthispaperweintroducetheStancutypegeneralizationoftheq-Bernstein-Schurer-Kantorovich
a
J operators and examine their approximation properties. We investigate the convergence of our oper-

5 ators with the help of the Korovkin’s approximation theorem and examine the convergence of these
2
operators in the Lipschitz class of functions. We also investigate the approximation process for these

] operators through the statistical Korovkin’s approximation theorem. Also, we present some direct
A
theorems for these operators. Finally we introduce the bivariate analogue of these operators and
C
study some results for the bivariate case.
.
h
at Keywords: Stancutypeq-Bernstein-Schurer-Kantorovichoperators;modulusofcontinuity;posi-
m tive linearoperators;Korovkintype approximationtheorem;statisticalapproximation,Lipschitz class
of functions.
[
AMS Subject Classifications (2010): 40A30, 41A10, 41A25, 41A36,
1
v
9 1 Introduction and preliminaries
1
3
6 The q-calculus has played an inportant role in the field of approximation theorey since last three
0 decades. In the year of 1987,A. Lupas was the first to apply the q-calculus to approximationtheory.
.
2 He introduced the q-analogue of the well known Bernstein polynomials [16]. Another remarkable ap-
0 plicationoftheq-calculusadventedintheyearof1997byPhillips[14]. Heusedtheq-calculustodefine
6
another interesting q-analogue of the classical Bernstein polynomials. Ostrovska [15] obtained more
1
resultsonthe q- Bernsteinpolynomials. Inthe sequelmanyresearchershavestudiedthe q-analogues
:
v of many well knownoperatorslike Baskakovoperators,Meyer-Konig-Zelleroperators,Szas-Mirakyan
i
X operators,Bleiman, Butzer and Hahn operators(written succinctly as BBH). Also the q-analoguesof
some integral operators like Kantorovich and Durrmeyer type were introduced and their approxima-
r
a tion properties were studied.
In [20] Muraru defined the q- Bernstein-Schurer operators in 2011. She used the modulus of conti-
nuity to obtain the rate of convergence of the q-Bernstein-Schurer operators. Recently, the Schurer
modifications of some positive linear operators have been studied in [1, 2, 3].
Kantorovich introduced the following integral type generalization of the classical Bernstein op-
eratos
n k+1
n+1
L (f;x)=(n+1) p (x) f(t)dt,
n n,k
k
Xk=0 Zn+1
where
n
p (x)= xk(1 x)n k, x [0,1],
n,k −
k − ∈
(cid:18) (cid:19)
is the Bernstein basis function. The approximation properties of these operators through the Ko-
rovkin’s approximation theorem were studied in [5].
1
Dalmanoglu presented another Kantorovich type generalization of the q-Bernstein polynomials and
studied some approximation results in [9]. Below we give some rudiments of the q-calculus. In [10],
Radu investigated the statistical convergence properties of the Bernstein-Kantorovich polynomials
basedon q-integers. Recently, many researchershave studied various q-extensions of the Kantorovich
operatos in [11, 12, 13].
For any fixed real number q >0 and k N 0 , the q-integer of k, denoted by [k] , is defined by
q
∈ ∪{ }
[k]q =( 11k−−,qqk, qq 6==11
and the q-factorial [k] ! is defined as
q
[k] [k 1] ...[1] , k 1
[k]q!= 1,q − q q k =≥0.
(cid:26)
The q- concept can be extended to any realnumber k. For integers n and k such that 0 k n, the
≤ ≤
q-analogue of the binomial coefficient is defined by
n [n] !
q
= .
k [k] ![n k] !
(cid:18) (cid:19)q q − q
For the q- binomial coefficient the following relations hold:
n n 1 n 1
= − +qk −
k k 1 k
(cid:18) (cid:19)q (cid:18) − (cid:19)q (cid:18) (cid:19)q
n n 1 n 1
= qn−k − + − .
k k 1 k
(cid:18) (cid:19)q (cid:18) − (cid:19)q (cid:18) (cid:19)q
Forx [0,1]andm N0,theq-analogueof(1+x)n,denotedby(1+x)n,isdefinedbythepolynomial
∈ ∈ q
(1+x)(1+qx)...(1+qn 1x), n=1,2,3,...
(1+x)n = −
q 1, n=0.
(cid:26)
For 0<q <1, a>0, the q-definite integral of a real valued function f is defined by
a
∞
f(x)d x=(1 q)a f(aqn)qn, a R,
q
− ∈
Z0 n=0
X
and over the interval [a,b], 0<a<b, it is defined by
b b a
f(x)d x= f(x)d x f(x)d x.
q q q
−
Za Z0 Z0
For details of these integrals one is referred to [1], [2]. In some situations the above integrals are
not appropriate to obtain the q-analogues of well known integrals. So we use another more general
integrals, the Reimann type q-integrals, defined as follows:
b
∞
f(x)dRx=(1 q)(b a) f(a+(b a)qs)qs,
q − − −
Za s=0
X
wherea,baresuchthat0 a<bandqisasabove. ThelaterintegralswereintroducedbyGauchman
≤
[3] and Marinkovic´et al.[4].
In this paper, let I denote the interval [0,1+l] equipped with the norm . , where l N0 =
C[0,1+l]
kk ∈
N 0 .
∪{ }
2
2 Construction of Operators
In 2015, P.N. Agarwal et al. [13] introduced the following Kantorovich type generalization of the
q-Bernstein-Schurer operators
n+p [k+1]q
Kn,p(f;q,x)=[n+1]q bqn+p,k(x)q−k [[nk]+q1]q f(t)dRqt, x∈[0,1], (2.1)
Xk=0 Z[n+1]q
wherebq (x)= n+p xk(1 x)n+p k istheq-Bernsteinbasisfunction. Theyhaveinvestigatedthe
n+p,k k q − q −
approximation properties of these operators using the Korovkin’s approximation theorem. Inspired
(cid:0) (cid:1)
by their work, we introduce the Stancu type generalisation of the Bernstein-Schurer-Kantorovich
operators based on q-integers as follows:
n+l [k+1]q+α
Lαn,,lβ(f;q;x)=([n+1]q+β) bkn,l(q;x)q−k [[kn]+q1+]αq+β f(t)dRqt, x∈[0,1], (2.2)
kX=0 Z[n+1]q+β
where bk (q;x) = n+l xk(1 x)n+l k is the q-Bernstein basis function and α,β are such that
n,l k q − q −
0<β α.
≤ (cid:0) (cid:1)
α = 0, β = 0 reduce the operators (2.2) to the operators (2.1). So the newly constructed
operators are a generalization of the operators in (2.1). We shall investigate some approximation
results for the operators in (2.2). To examine the approximation results, we need the following
lemmas.
Lemma 2.1 Let Lα,β(f;q;x) be given by (2.1). Then the followings hold:
n,l
(i) Lα,β(1;q;x)=1,
n,l
(ii) Lα,β(t;q;x)= α + 1 + 2q[n+l]q x,
n,l [n+l]q ([n+1]q+β)[2]q ([n+1]q+β)[2]q
(iii) Lα,β(t2;q;x)= 1 + 2α + α2 + q[n+l]q((3+4α)+(5+4α)q+4(1+α)q2)x
n,l ([n+1]q+β)2[3]q ([n+1]q+β)2[2]q ([n+1]q+β)2 ([n+1]q+β)2[2]q[3]q
+ q2[n+l]q[n+l−1]q(1+q+4q2)x2.
([n+1]q+β)2[2]q[3]q
Proof. Before proving the above lemma, we shall first prove the following:
n+l
bk (q;x)qk =1 (1 q)[n+l] x, (2.3)
n,l − − q
k=0
X
and
n+l
bk (q;x)q2k =1 (1 q2)[n+l] x+q(1 q)2[n+l][n+l 1] x2 (2.4)
n,l − − q − − q
k=0
X
where bk (q;x)= n+l xk(1 x)n+l k.
n,l k q − q −
(cid:0) (cid:1)
3
In fact we have
n+l n+l
bk (q;x)qk = bk (q;x) 1 1+q)[k]
n,l n,l − q
k=0 k=0
X X (cid:0) (cid:1)
n+l n+l
[k]
= bk (q;x) (1 q)[n+l] bk (q;x) q
n,l − − q n,l [n+l]
q
k=0 k=0
X X
n+l 1
− n+l 1
= 1−(1−q)[n+l]q k− xk+1(1−x)qn+l−k−1
k=0 (cid:18) (cid:19)q
X
= 1 (1 q)[n+l] x,
q
− −
and
n+l n+l
bk (q;x)q2k = bk (q;x) (1 1+q2)[k] +q(1 q)2[k 1] [k]
n,l n,l − q − − q q
k=0 k=0 (cid:18) (cid:19)
X X
n+l n+l
[k]
= bk (q;x) (1 q2)[n+l] bk (q;x) q +q(1 q)2[n+l] [n+l 1]
n,l − − q n,l [n+l] − q − q
q
k=0 k=0
X X
n+l
[k] [k 1]
bk (q;x) q − q
× n,l [n+l] [n 1]
q q
k=0 −
X
n+l
n+l 1
= 1−(1−q2)[n+l]q k− xkx(1−x)qn+l−k−1+q(1−q)2[n+l]q[n+l−1]q
k=0(cid:18) (cid:19)q
X
n+l 2
− n+l 2
× k− xkx2(1−x)qn+l−k−2.
k=0 (cid:18) (cid:19)q
X
(i)
n+l [k+1]q+α
L α,β(1;q;x) = ([n+1] +β) bk (q;x)q k [n+1]q+β 1dRt
n,l q n,l − q
[k]q+α
kX=0 Z[n+1]q+β
n+l
= ([n+1]q+β) bkn,l(q;x)q−k(1−q)([(k[n++11]q]−+[kβ]q)) ∞ qs
q
k=0 s=0
X X
n+l
= bk (q;x)=1.
n,l
k=0
X
(ii)
n+l [k+1]q+α
L α,β(t;q;x) = ([n+1] +β) bk (q;x)q k [n+1]q+β tdRt
n,l q n,l − q
[k]q+α
kX=0 Z[n+1]q+β
n+l
= ([n+1]q+β) bkn,l(q;x)q−k(1−q)([(k[n++11]q]−+[kβ])q) ∞ f [n[+k]q1]++αβ + [k[n++11]q]−+[kβ]qqs qs
k=0 q s=0 (cid:18) q q (cid:19)
X X
= n+lbk (q;x)(1 q) ∞ [k]q+α + qkqs qs
n,l − [n+1] +β [n+1] +β
k=0 s=0(cid:18) q q (cid:19)
X X
n+l [k] +α qk 1
= bk (q;x) q +
n,l ([n+1] +β) ([n+1] +β)[2]
k=0 (cid:18) q q q(cid:19)
X
4
n+l
[n+l] [k] +α 1
= q bk (q;x) q + (1 (1 q)[n+p] x)
[n+1] +β n,l [n+l] [2] ([n+1] +β) − − q
q q q q
k=0
X
n+l n+l
[n+l] [k] α 1 (1 q)[n+l] x
= q bk (q;x) q + bk (q;x) + − − q
[n+1] +β n,l [n+l] n,p [n+l] [2] ([n+1] +β)
q " q q# q q
k=0 k=0
X X
n+l
[n+l] n+l [k] α 1 (1 q)[n+l] x
= q xk(1 x)n+l−k q + + − − q
[n+1]q+β"k=0(cid:18) k (cid:19)q − [n+l]q [n+l]q# [2]q([n+1]q+β)
X
n+l 1
= [n+l]q − n+l−1 xk(1 x)n+l−k−1+ α + 1−(1−q)[n+l]qx
[n+1]q+β" k=1 (cid:18) k−1 (cid:19)q − [n+l]q# [2]q([n+1]q+β)
X
[n+l] α 1 (1 q)[n+l] x
q q
= x+ + −
[n+1] +β [n+l] [2] ([n+1] +β) − [2] ([n+1] +β)
q q q q q q
[n+l] (1 q)[n+l] α 1
q q
= − x+ +
"[n+1]q+β − [2]q([n+1]q+β)# [n+l]q [2]q([n+1]q+β)
α 1 2q[n+l]
q
= + + x.
[n+l] [2] ([n+1] +β) [2] ([n+1] +β)
q q q q q
(iii)
n+l 2
L αn,,lβ(t2;q;x) = ([n+1]q+β) bkn,l(q;x)q−k[(k[n++11]q]−+[kβ]q) ∞ [n[+k]q1]++αβ + [k[n++11]q]−+[kβ]qqs qs
q q q !
k=0 s=0
X X
= n+lbk (q;x)(1 q) ∞ ([k]q+α)2 + q2kq2s + 2qkqs([k]q+α) qs
n,l − ([n+1]q+β)2 ([n+1]q+β)2 ([n+1]q+β)2!
k=0 s=0
X X
n+l ([k] +α)2 2qk([k] +α) q2k
= bk (q;x) q + q +
n,l ([n+1] +β)2 ([n+1] +β)2(1+q) ([n+1] +β)2(1+q+q2)
q q q !
k=0
X
n+l [k]2 n+l 2α[k] n+l α2
= bk (q;x) q + bk (q;x) q + bk (q;x)
n,l ([n+1] +β)2 n,l ([n+1] +β)2 n,p ([n+1] +β)2
q q q
k=0 k=0 k=0
X X X
n+l 2qk[k] n+l 2αqk
+ bk (q;x) q + bk (q;x)
n,l ([n+1] +β)2[2] n,l ([n+1] +β)2[2]
q q q q
k=0 k=0
X X
n+l q2k
+ bk (q;x)
n,l ([n+1] +β)2[3]
q q
k=0
X
[n+l] 1 [n+l 1] 2α[n+l] α2
q q q
= x +q − x ++ x+
[n+1] +β [n+1] +β [n+1] +β ([n+1] +β)2 ([n+1] +β)2
q q q ! q q
2q[n+l] 2q(1 q)[n+l] [n+l 1] 2α
+ q x − q − qx2+
[2] ([n+1] +β)2 − [2] ([n+1] +β)2 [2] ([n+1] +β)2
q q q q q q
1
1 (1 q)[n+l] x +
× − − q [3] ([n+1] +β)2
(cid:18) (cid:19) q q
1 (1 q2)[n+l] x+q(1 q)2[n+l] [n+l 1] x2
q q q
× − − − − !
5
1 2α α2 [n+l] 2α[n+l]
q q
= + + + +
[3] ([n+1] +β)2 [2] ([n+1] +β)2 ([n+1] +β)2 ([n+1] +β)2 ([n+1] +β)2
q q q q q (cid:20) q q
2q[n+l] 2α(1 q)[n+l] (1 q2)[n+l] q[n+l] [n+l 1]
q q q q q
+ − − x+ −
[2] ([n+1] +β)2 − [2] ([n+1] +β)2 − [3] ([n+1] +β)2 ([n+1] +β)2
q q q q q q (cid:21) (cid:20) q
2q(1 q)[n+l] [n+l 1] q(1 q)2[n+l] [n+l 1]
− q − q + − q − q x2
− [2] ([n+1] +β)2 [3] ([n+1] +β)2
q q q q (cid:21)
1 2α α2 [n+l]
q
= + + +
[3] ([n+1] +β)2 [2] ([n+1] +β)2 ([n+1] +β)2 ([n+1] +β)2[2] [3]
q q q q q q q q
q[n+l] [n+l 1]
[2] [3] +2α[2] [3] +2q[3] 2α(1 q)[3] (1 q2)[2] x+ q − q
× q q q q q− − q− − q ([n+1] +β)2[2] [3]
(cid:20) (cid:21) q q q
[2] [3] 2(1 q)[3] +(1 q)2[2] x2
q q q q
× − − −
(cid:20) (cid:21)
1 2α α2 [n+l]
q
= + + +
[3] ([n+1] +β)2 [2] ([n+1] +β)2 ([n+1] +β)2 ([n+1] +β)2[2] [3]
q q q q q q q q
q[n+l] [n+l 1]
q (1+q)(1+2q)+(1+q+q2)(4α+2) x+ q − q
× ([n+1] +β)2[2] [3]
(cid:18) (cid:19) q q q
(1+q+q2)(3q 1)+(1+q2 2q)(1+q) x2
× − −
(cid:18) (cid:19)
1 2α α2
= + +
[3] ([n+1] +β)2 [2] ([n+1] +β)2 ([n+1] +β)2
q q q q q
+q[n+l]q (3+4α)+(5+4α)q+4(1+α)q2 x+ q2[n+l]q[n+l−1]q(1+q+4q2)x2.
([n+1] +β)2[2] [3] ([n+1] +β)2[2] [3]
(cid:0) q q q (cid:1) q q q
Hence the lemma.
Remark 2.1. From the Lemma 2.1, we have
(i) Lα,β((t x);q;x)= 2q[n+l]q 1 x+ 1 + α ,
n,l − [2]q([n+1]q+β − [2]q([n+1]q+β) [n+l]q
(cid:18) (cid:19)
(ii) Lα,β((t x)2;q;x)= α2 + 2α + 1
n,l − ([n+1]q+β)2 ([n+1]q+β)2[2]q ([n+1]q+β)2[3]q
+ q[n+l]q (3+4α)+(5+4α)q+4(1+α)q2 2 2α x
([n+1]q+β)2[2]q[3]q − ([n+1]q+β)2[2]q − [n+l]q
(cid:18) (cid:0) (cid:1) (cid:19)
+ q2[n+l]q[n+l−1]q(1q+4q2) 4q[n+l]q +1 x2.
[n+1]q+β)2[2]q[3]q − [n+1]q+β)[2]q
(cid:18) (cid:19)
Lemma 2.2 For any f C(I), we have Lα,β(f;q;.) f .
∈ k n,l kC[0,1] ≤k kC[0,1+l]
3 Direct Theorems
In this section, we prove some direct theorems for the operators Lα,β(f;q;x).
n,l
Theorem 3.1 Let f C(I) and 0 < q < 1. Then the sequence of the operators Lα,β(f;q ;.)
∈ n n,l n
converges uniformly to f on the compact interval [0,1] if and only if lim q =1.
n
n
→∞
Proof. (Forward) Suppose that lim q = 1. Then we shall show that Lα,β(f;q ;.) converges to
n n n,l n
f uniformly on [0,1]. Note that fo→r∞0 < q < 1 and q for n , we get [n+1]
n n → ∞ → ∞ qn → ∞
6
as n → ∞. Now it is easily seen that [n[+n1+]ql]nqn+β = 1+qnn([[nl]+qn1−]qn1) − [n+1β]qn+β. So when n → ∞,
[n+l]qn 1 and [n+l]qn 0. Using this and the Lemma 2.1, we find that Lα,β(1;q ;x) 1,
[n+1]qn+β → ([n+1]qn+β)2 → n,l n →
Lα,β(t;q ;x) x and Lα,β(t2;q ;x) x2 uniformly on the compact set [0,1] as n . Therefore,
n,l n → n,l n → →∞
the Korovkin’s theorem proves that the sequence Lα,β(f;q ;.) converges uniformly to f on [0,1].
n,l n
We shallprovethe converseby the method ofcontradiction. Suppose thatthe sequence (q ) doesnot
n
converge to 1. Then there must exist a subsequence (q ) of the sequence (q ) such that q (0,1),
qni → δ ∈ [0,1) as i → ∞. Then [ni+1p]qni = 1−(1q−niq)nnniii+p → 1−δ as i →n∞ because (qnnii)∈ni → 0
as i . Now if we choose n = n ,q = q in Lα,β(t;q;x) form the Lemma 2.1, then we get
→ ∞ i ni n,l
Lα,β(t;q;x) = 2δ x+ 1 δ 1 +α(1 δ), which is different from x when i ,
n,l (1+δ)(1+β(1 δ)) 1+−δ(1+β(1 δ)) − → ∞
which contradicts our sup−position. Hence li−m q =1. Hence the theorem is completely proved.
n
n
→∞
Now we define the following:
Let f C(I), δ >0 and W2 = h:h,h C(I) , then the Peetre’s K-functional is defined by
′ ′′
∈ { ∈ }
K (f,δ)= inf f h +δ h ,
2 ′′
h W2{k − k k k}
∈
By DeVore and Lorentz theorem [?]there exists a constant C >0 such that
K (f,δ) Cω (f,√δ) (3.1)
2 2
≤
where ω (f,√δ), the second order modulus of continuity of f C(I), is defined as
2
∈
ω (f,√δ)= sup sup f(x+2p) 2f(x+p)+f(x).
2
| − |
1 x I
0<p<δ2 ∈
Also by ω(f,δ), we denote the first order modulus of continuity of f C(I) defined as
∈
ω(f,δ)= sup sup f(x+p) f(x)
| − |
0<p<δx I
∈
Next we prove the following theorem.
Theorem 3.2 Let Lα,β(f;q;x) be the sequence of positive linear operators defined by (2.1) and f
n,l ∈
C(I). Let (q ) be the sequence with 0<q < 1 and q 1 as n . Then there exists a constant
n n n
→ →∞
c>0 indepenent of n and x such that
α 1 2q [n+l]
Lα,β(f;q;x) f(x) Cω f, φα,β(q ;x) +ω f, + + n qn x x
n,l − ≤ 2 n,l n [n+l] [2] ([n+1] +β) [2] ([n+1] +β) −
(cid:18) q (cid:19) (cid:18) qn qn qn qn qn (cid:19)
(cid:12) (cid:12) (3.2)
(cid:12) (cid:12)
2
where φαn,,lβ(qn;x) = Lαn,,lβ((t − x)2;qn;x) + [n+αl]qn + [2]qn([n+11]qn+β) + [2]q2nq(n[n[n++1l]q]qnn+β)x−x and
x [0,1]. (cid:16) (cid:17)
∈
Proof. Let us define the following operators
α 1 2q [n+l]
L¯α,β(f;q ;x)=Lα,β(f;q ;x)+f(x) f + + n qn x
n,l n n,l n − [n+l] [2] ([n+1] +β) [2] ([n+1] +β)
(cid:18) qn qn qn qn qn (cid:19)
(3.3)
In the light of the Lemma 2.1, it is easily seen that L¯α,β(1;q ;x) = 1 and L¯α,β(t;q ;x) = x. Now
n,l n n,l n
from the Taylor’s formula, for g W2, we can write
∈
t
g(t) g(x)=(t x)g (x)+ (t u)g (u)du
′ ′′
− − −
Zx
7
Applying the operators L¯α,β to both sides of the above equation we get
n,l
t
L¯α,β(g;q ;x) g(x) = g (x)L¯α,β((t x);q ;x)+L¯α,β (t u)g (u)du
n,l n − ′ n,l − n n,l − ′′
(cid:18) Zx (cid:19)
t
= L¯αn,,lβ (t−u)g′′(u)du,qn;x
(cid:18) Zx (cid:19)
t
= Lα,β (t u)g (u)du,q ;x
n,l − ′′ n
(cid:18) Zx (cid:19)
[n+αl]qn+[2]qn([n+11]qn+β)+[2]q2nq(n[n[n++1l]]qqnn+β)x α 1
+
− [n+l] [2] ([n+1] +β)
Zx (cid:18) qn qn qn
2q [n+l]
+ n qn x u g (u)du
′′
[2] ([n+1] +β) −
qn qn (cid:19)
Therefore, we will have
α 1
L¯α,β(g;q ;x) g(x) Lα,β((t x)2;q ;x) g + +
| n,l n − | ≤ n,l − n k ′′kC[0,1+l] [n+l] [2] ([n+1] +β)
(cid:18) qn qn qn
2q [n+l] 2
+[2] ([nn+1] qn+β)x−x kg′′kC[0,1+l]
qn qn (cid:19)
= φα,β(q ;x) g
n,l n k ′′kC[0,1+l]
In view of (3.2), we obtain
Lα,β(f;q ;x) f(x) L¯α,β(f g;q ;x) g(x) + L¯α,β(g;q ;x) g(x)
| n,l n − | ≤ | n,l − n − | | n,l n − |
α 1 2q [n+l]
+ f + + n qn x f(x)
[n+l] [2] ([n+1] +β) [2] ([n+1] +β) −
(cid:12) (cid:18) qn qn qn qn qn (cid:19) (cid:12)
(cid:12) (cid:12)
Now we have (cid:12) (cid:12)
(cid:12) (cid:12)
L¯α,β(f;q ;x) 3 f ,
k n,l n k≤ k kC[0,1+l]
by the Lemma 2.2, so we have
Lα,β(f;q ;x) f(x) 4 f g +φα,β(q ;x) g +ω f, 1
| n,l n − |≤ k − kC[0,1+l] n,l n k ′′kC[0,1+l] ([n+1]qn+β)[2]qn
(cid:18) (cid:12)
(cid:12)
+ [n+l]qn 2qn x x . (cid:12)
([n+1]qn+β)[2]qn − (cid:12)
(cid:12)(cid:19)
On taking the infimum of the right hand side runn(cid:12)ing over all g W2 and using the definition of the
(cid:12) ∈
Peetre’s functional, we get (cid:12)
1 [n+l] 2q
Lα,β(f;q ;x) f(x) 4K (f,φα,β(q ;x))+ω f, + qn n x x
| n,l n − |≤ 2 n,l n ([n+1] +β)[2] ([n+1] +β)[2] −
(cid:18) qn qn qn qn (cid:12)(cid:19)
(cid:12)
Now in view of (3.1), we obtain (cid:12)
(cid:12)
1 [n+l] 2q
Lα,β(f;q ;x) f(x) Cω f, φα,β(q ;x) +ω f, + qn n x x ,
| n,l n − |≤ 2 n,l n ([n+1] +β)[2] ([n+1] +β)[2] −
(cid:18) q (cid:19) (cid:18) (cid:12) qn qn qn qn (cid:12)(cid:19)
(cid:12) (cid:12)
and this completes the proof of the theorem. (cid:12) (cid:12)
(cid:12) (cid:12)
Now we shall obtain an estimate of the rate of convergence for the operators defined in (2.1) using
the Lipschitz-type maximal function defined as follows [?]
For x [0,1] and ξ (0,1], the Lipschitz-type maximal function is defined as
∈ ∈
f(t) f(x)
ω˜ (f,x)= sup | − | (3.4)
ξ t xξ
t6=x,t∈[0,1+l] | − |
8
Now we prove the following theorem
Theorem 3.3 Let f C(I),0 < ξ 1 and q (0,1) such that q 1 as n . Then for every
n n
∈ ≤ ∈ → →∞
x [0,1], we have
∈
|Lαn,,lβ(f;qn;x)−f(x)|≤ω˜ξ(f,x)(γn,l(qn;x))ξ2,
where γ (q ;x)=Lα,β((t x)2;q ;x)
n,l n n,l − n
Proof. In the light of the Lemma (2.1), we have
Lα,β(f;q ;x) f(x) Lα,β(f(t) f(x);q ;x) ω˜ (f,x)Lα,β(t xξ;q ;x) (3.5)
| n,l n − |≤ n,l | − | n ≤ ξ n,l | − | n
and in view of (3.4), we have
f(t) f(x) ω˜ (f,x)t xξ. (3.6)
ξ
| − |≤ | − |
When we use the Ho¨lder’s inequality with p˜= 2 and q˜= 2 , we obtain
ξ 2 ξ
−
|Lαn,,lβ(f;qn;x)−f(x)|≤ω˜ξ(f,x)Lαn,,lβ(|t−x|2;qn;x)ξ2 =ω˜ξ(f,x)(γn,l(qn;x))ξ2,
and hence the theorem.
To prove the next theorem we consider the following Lipschitz-type space of functions [?]:
t xs
L˜ip (s)= f C(I): f(t) f(x) M | − | ,
M ∈ | − |≤ (t+x)s2
(cid:26) (cid:27)
where M is a positive constant and 0<s 1.
≤
Now we have the following theorem.
Theorem 3.4 Let f L˜ip (s),s (0,1] and q (0,1) such that q 1 as n . Then for each
∈ M ∈ n ∈ n → →∞
x (0,1], we have
∈ s
Lα,β(f;q ;x) f(x) M γn,l(qn;x) 2
| n,l n − |≤ x
(cid:18) (cid:19)
where γ (q ;x)=Lα,β((t x)2;q ;x).
n,l n n,l − n
Proof. Firstly we will prove the result for s=1. In fact we have, for f L˜ip (1),
∈ M
n+l [k+1]qn+α
Lα,β(f;q ;x) f(x) ([n+1] +β) bk (q ;x)q k [n+1]qn+β f(t) f(x)dR t
| n,l n − | ≤ qn n,l n n− [k]qn+α | − | qn
kX=0 Z[n+1]qn+β
≤ M([n+1]qn +β)n+lbkn,l(qn;x)qn−k [[[knk]++qn11]]+qqnnα++αβ √|tt−+xx|dRqnt
kX=0 Z[n+1]qn+β
Applying the Cauchy-Schwarzinequality and the fact 1 1 , we get
√t+x ≤ √x
|Lαn,,lβ(f;qn;x)−f(x)| ≤ √Mx([n+1]qn +β)n+lbkn,l(qn;x)qn−k [[[knk]++qn11]]+qqnnα++αβ |t−x|dRqnt
kX=0 Z[n+1]qn+β
M
= Lα,β(t x;q ;x)
√x n,l | − | n
1
γn,l(qn;x) 2
M .
≤ x
(cid:18) (cid:19)
9
Thustheresultisestablishedfors=1. Nextweprovetheresultfor0<s<1. OnusingtheHo¨lder’s
inequality twice for p˜= 1 and q˜= 1 , we get
s 1 s
−
n+l [k+1]qn+α
|Lαn,,lβ(f;qn;x)−f(x)| ≤ ([n+1]qn +β) bkn,l(qn;x)qn−k [[kn]+qn1]+qnα+β |f(t)−f(x)|dRqnt
kX=0 Z[n+1]qn+β
n+l [k+1]qn+α s1 s
bk (q ;x) ([n+1] +β)q k [n+1]qn+β f(t) f(x)dR t
≤ ( n,l n qn n− [k]qn+α | − | qn ! )
Xk=0 Z[n+1]qn+β
n+l [k+1]qn+α s
≤ (([n+1]qn +β) bkn,l(qn;x)qn−k [[kn]+qn1]+qnα+β |f(t)−f(x)|1sdRqnt) .
kX=0 Z[n+1]qn+β
Now as f L˜ip (s), we obtain
∈ M
|Lαn,,lβ(f;qn;x)−f(x)| ≤ M(([n+1]qn +β)n+lbkn,l(qn;x)qn−k [[[knk]++qn11]]+qqnnα++αβ √|tt−+xx|dRqnt)s
kX=0 Z[n+1]qn+β
M ([n+1] +β)n+lbk (q ;x)q k [[nk++11]]qqnn++αβ t xdR t s
≤ x2s ( qn n,l n n− [k]qn+α | − | qn )
Xk=0 Z[n+1]qn+β
M
= (Lα,β(t x;q ;x))s
x2s n,l | − | n
s
γn,l(qn;x) 2
M .
≤ x
(cid:18) (cid:19)
This completes the proof of the theorem.
4 Statistical Convergence
In this section we study the A-statistical convergence of the operators defined in (2.1) through the
Korovkin-type statistical approximation theorem.
Let A = (a ) be a non-negative infinite summability matrix. Then for a sequence x = (x ), we
nk k
define the A-transform of x as (Ax)n = ∞k=1 whenever the series converges for each n. We denote
it by Ax=((Ax) ). A is said to be regular if lim (Ax) =L, whenever lim (x) =L. The sequence
n n n n n
P
x = (x) is said to be A-statistically convergent to the limit L, denoted by st lim x = L, if for
n A n n
−
each ǫ > 0,lim a = 0. If we take the matrix A to be the Cesa`ro matrix C of order
one, then the nnotiokn:|oxkf−tLh|e>Aǫ -sntkatistically convergence is same as the statistical convergence.
P
Now to prove a theorem, we take a sequence (q) such that q (0,1) satisfying the following:
n n
∈
st lim x =1, st lim (q )n =a (0,1), st lim 1 =0.
A− n n A− n n ∈ A− n [n]qn
Theorem 4.1 Let A = (a ) be a non-negative regular summability matrix and (q ) be a sequence
nk n
satisfying the above conditions. Then for any f C(I), we have
∈
st lim Lα,β(f;q;.) f =0.
A− n k n,l − kC[0,1]
Proof. Let e (x)=xi, where x [0,1], i=0,1,2. Then from the Lemma 2.1, we have
i
∈
st lim Lα,β(e ;q;.) e =0. (4.1)
A− n k n,l 0 − 0kC[0,1]
Next, again from the Lemma 2.1, we have
α 1 2q[n+l]
lim Lα,β(e ;q ;.) e + + q 1 .
n k n,l 1 n − 1kC[0,1] ≤(cid:12)[n+l]q ([n+1]q+β)[2]q(cid:12) (cid:12)([n+1]q+β)[2]q − (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) 10 (cid:12) (cid:12) (cid:12)