6.On Cheney-Sharma Chlodovsky Operators

ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 11 Issue 1(2019), Pages 36-43.

ON CHENEY-SHARMA CHLODOVSKY OPERATORS

DILEK S ¨OYLEMEZ, FATMA TAS¸DELEN

Abstract. The main purpose of this paper is to show that Cheney-Sharma Chlodovsky operators preserve properties of the function of modulus of con-tinuity and Lipschitz condition of a given Lipschitz continuous function f.

Furthermore, we give a result for these operators whenfis a convex function.

1. _Introduction

The Bernstein polynomialsBn :C[0,1]→C[0,1] are defined by n

X

k=0 f

_k n

n k

xk(1−x)n−k, x∈[0,1], n∈N.

In 1982, Lindvall [21] proved that Bernstein polynomials preserve the Lipschitz constant and order of a Lipschitz continuous function using a probabilistic method. Later, Brown, Elliott and Paget [7] gave an elementary proof for the same result. Furthermore, using similar method as in [7], Li [20] showed that Bernstein poly-nomials preserve the properties of the function of modulus of continuity. These problems were investigated for some linear positive operators via elementary meth-ods or probabilistic methmeth-ods (see, for instance [3], [4], [5], [6], [12], [13] [18], [19], [28] ).

In [10], Chlodovsky introduced the classical Bernstein-Chlodovsky polynomials as a generalization of Bernstein polynomials on unbounded sets. For everyn∈_N,

f ∈C[0,∞) and x∈ [0,∞), these polynomialsCn : C[0,∞)→C[0,∞) defined by

Cn(f, x) :=  

 n P k=0

f k_nbn n

k x

bn k

1− x bn

n−k

, if 0≤x≤bn

f(x) , ifx > bn

, (1.1)

where 0≤x≤bn and{bn}is a positive sequence with properties: limn→∞bn=∞, limn→∞(bn/n) = 0.

Recently, some authors have studied some Chlodovsky type polynomials which may be found in ( [4], [8], [14], [15], [16], [17], [22]).

2000Mathematics Subject Classification. 41A36.

Key words and phrases. Lipschitz continuous function; Cheney-Sharma Chlodovsky operators; modulus of continuity function.

c

2019 Universiteti i Prishtin¨es, Prishtin¨e, Kosov¨e. Submitted October 18, 2018. Published February 18, 2019. Communicated by F. Marcellan.

It is known that the Abel-Jensen equalities are given by the following formulas (see [2], p.326)

(u+v+nβ)n−1= n X

k=0 _n

k

u(u+kβ)k−1v[v+ (n−k)β]n−k−1 (1.2) and

(u+v) (u+v+mβ)m−1= m X

k=0 _m

k

u(u+kβ)k−1v[v+ (m−k)β]m−k−1,

(1.3) whereu,vandβ∈R. By means of these equalities Cheney-Sharma [9], introduced the following Bernstein type operators, forf ∈C[0,1],x∈[0,1] andn∈N,

Qn(f;x) = (1 +nβ)1− n

n X

k=0 f

k n

n k

x(x+kβ)k−1t_nk,0(x,1)n−k (1.4) and

Gn(f;x) = (1 +nβ)1−n n X

k=0 f

_k n

n k

x(x+kβ)k−1(1−x)

tk,_n0(x,1)n−k−1 ,

(1.5) whereβ is a nonnegative real parameter and the notation

tk,jn (x, z) = 1−

x

z + (n−k−j)β

is used for convenience. It is obvious that forβ = 0 these operators reduce to the classical Bernstein operators. Cheney-Sharma proved that ifnβn→ ∞as n→ ∞, then forf ∈ C[0,1], these operators uniformly converge to f on [0,1]. In [5], the authors showed that Cheney-Sharma operators preserve the Lipschitz constant and order of a Lipschitz continuous function as well as the properties of the function of modulus of continuity. They also gave a result forGn(f;x) under the convexity of

f. A Kantorovich type generalization of the Cheney-Sharma operators was studied in [23]. For these operators, we refer the readers to ([1], [6], [11], [24], [26], [27]).

In [25],Chlodovsky-type Cheney-Sharma operators were defined as

G∗_n(f;x) = (1 +nβ)1−n n X

k=0 f

_k nbn

n k

_x bn

_x bn

+kβ k−1

(1.6)

×

1− x

bn

tk,n0(x, bn) n−k−1

,

for 0≤x≤bnandf(x) forx > bn and{bn}satisfies the conditions given by (1.1). The authors also proved the weighted uniform convergence ofG∗_n(f)→f, obtained the rate of approximation in terms of the usual modulus of continuity and studied

A-statistical convergence behaviours of these operators. For the case ofbn= 1 the operatorsG∗n(f;x) reduce to Gn(f;x).

Also, from Lemma 2, in [25],the operatorsG∗

n(f;x) satisfy

G∗_n(1;x) = 1, (1.7)

G∗_n(t;x) =x. (1.8)

and Lipschitz condition of a given Lipschitz continuous functionf. Furthermore, we prove that

G∗_n(f)≥f

for all convex functions f ∈ C[0,∞). In order to give some properties of the operators, defined by (1.6),let us recall some useful definitions.

Definition 1. If a continuous, non-negative function,ω,defined on[0, A],satisfies the conditions:

1. ω(u+v) ≤ ω(u) +ω(v) for u, v, u+v ∈ [0, A] (A > 0), i.e. ω is semi-additive,

2. ω(u)≥ω(v)foru≥v,i.e. ω is non-decreasing,

3. limu→0+ω(u) =ω(0) = 0,then it is called a function of modulus of

continu-ity.

Definition 2. Let f be a real valued continuous function defined on[0,∞).Then

f is said to be Lipschitz continuous of orderγ on [0,∞)if the following inequality holds

|f(x)−f(y)| ≤M|x−y|γ

for x, y ∈ [0,∞) with M > 0 and 0 < γ ≤ 1. The set of Lipschitz continuous functions is denoted by LipM(γ).

Definition 3. A real valued continuous functionf is said to be convex on[0,∞),

if the following inequality holds

f

n X

k=1 αkxk

!

≤ n X

k=1

αkf(xk)

for allx1,x2,..., xn∈[0,∞)and for all non-negative numbers,α1, α2, ...αn such that

α1+α2+...+αn = 1.

2. Main Results

In this section, we prove that Cheney-Sharma Chlodovsky operators preserve properties of the function of modulus of continuity when the attached functionf is a function of modulus of continuity and the Lipschitz condition of a given Lipschitz continuous functionf,as in [7] and [20]. Finally, we give an inequality forG∗_n(f;x), using the convexity off.

Theorem 1. If ω is a function of modulus of continuity then G∗_n(ω;x)is also a function of modulus of continuity.

Proof. Letx, y∈[0, bn) andy≥x

G∗_n(ω;y) = (1 +nβ)1−n n X

j=0 ω

_j nbn

n j

_y bn

_y bn

+jβ j−1

(2.1)

×

1− y

bn

tj,_n0(y, bn) n−j−1

.

Lettingu=x/bn, v= (y−x)/bn andn=j in (1.3), we obtain

y bn

_y bn

+jβ j−1

= j X

k=0 _j

k _x

bn _x

bn +kβ

k−1_y−x bn

_y−x bn

and, with use of (2.1), we have

G∗_n(ω;y) = (1 +nβ)1−n n X

j=0

j X

k=0 ω

_j nbn

n j

_j k

_x bn

_x bn

+kβ k−1

×y−x

bn _y

−x bn

+ (j−k)β

j−k−1

1− y

bn

tj,n0(y, bn) n−j−1

.

In the last equality, let us change the order of the summations as follows:

G∗n(ω;y) = (1 +nβ)1−n

n X

k=0

n X

j=k

ω _j

nbn

_n!

(n−j)!j!

j!

k! (j−k)!

x bn

_x bn

+kβ k−1

×y−x

bn

y−x bn

+ (j−k)β

j−k−1

1− y

bn

tj,n0(y, bn) n−j−1

. (2.2)

Now, takingj−k=lin (2.2), we obtain

G∗_n(ω;y) = (1 +nβ)1−n

n X

k=0

n−k X

l=0 ω

_k+l n bn

_n!

k!l! (n−k−l)!

x bn

_x bn

+kβ k−1

×y−x

bn

_y−x bn

+lβ l−1

1− y

bn

tk,l_n (y, bn)

n−k−l−1

. (2.3)

In (1.3), replace u, v and n by (y−x)/bn, 1−y/bn and n−k, respectively, to obtain

1− x

bn

tk,_n0(x, bn) n−k−1

= n−k X

l=0

n−k l

y−x bn

y−x bn

+lβ l−1

1− y

bn

tk,l_n (y, bn)

n−k−l−1 ,

which implies

G∗_n(ω;x) = (1 +nβ)1−n

n X

k=0

n−k X

l=0 ω

_k nbn

n−k l

_n

k _x

bn _x

bn +kβ

k−1

×

_y−x bn

y−x bn

+lβ l−1

1− y

bn

tk,l_n (y, bn)

n−k−l−1

According to the equations (2.3) and (2.4) one has

G∗_n(ω;y)−G∗_n(ω;x) = (1 +nβ)1−n

n X

k=0

n−k X

l=0

ω

k+l n bn

−ω

k nbn

n!

k!l! (n−k−l)! × x

bn _x

bn +kβ

k−1_y−x bn

y−x bn

+lβ l−1

1− y

bn

tk,l_n (y, bn)

n−k−l−1 .

(2.5)

By Definition 1, we have

ω _k+l

n bn

−ω _k

nbn

≤ω _l

nbn

,

therefore,

G∗_n(ω;y)−G∗_n(ω;x) ≤(1 +nβ)1−n

n X

k=0

n−k X

l=0 ω

l nbn

n!

k!l! (n−k−l)!

x bn x bn +kβ k−1 ×

_y−x bn

y−x bn

+lβ l−1

1− y

bn

tk,l_n (y, bn)

n−k−l−1 .

By changing the order of the summations, we have

G∗_n(ω;y)−G∗_n(ω;x) ≤(1 +nβ)1−n

n X

l=0

n−l X

k=0 ω

_l nbn

_{n! (n−l)!} l! (n−l)!k! (n−k−l)!

x bn _x bn +kβ k−1 × y−x

bn

y−x bn

+lβ l−1

1− y

bn

tk,ln (y, bn)

n−k−l−1

(2.6)

= (1 +nβ)1−n n X

l=0 ω

_l nbn

n l

y−x bn

y−x bn

+lβ l−1

× n−l X

k=0 _n−l

k _x bn _x bn +kβ k−1

1− y

bn

tk,l_n (y, bn)

n−k−l−1 .

In (1.3), if we takeu, v and n, byx/bn,1−y/bn and n−l, respectively, then we have

1−(y−x)

bn

tl,_n0(y−x, bn) n−l−1

= n−l X

k=0 _n−l

k x bn x bn +kβ k−1

1− y

bn

tk,l_n (y, bn)

Rearranging the equality (2.6),we obtain

G∗_n(ω;y)−G∗_n(ω;x) ≤(1 +nβ)1−n

n X

l=0 ω

_l nbn

n l

y−x bn

y−x bn

+lβ l−1

×

1−(y−x)

bn

tl,_n0(y−x, bn) n−l−1

=G∗_n(ω;y−x).

Hence{G∗_n}is semi-additive. We also infer from (2.5) that{G∗_n}is non decreasing. From the definition of{G∗_n}it is clear that

lim x→0G

∗

n(ω) (x) =G

∗

n(ω) (0) =ω(0) = 0.

So the proof is completed.

Theorem 2. If f ∈LipM(γ),then Gn∗(f;x)∈LipM(γ).

Proof. Letx, y∈[0, bn) andy≥x.According to (2.5) we can write |G∗_n(f;y)−G∗_n(f;x)|

≤(1 +nβ)1−n n X

k=0

n−k X

l=0 f

_k+l n bn

−f _k

nbn

n!

k!l! (n−k−l)!

x bn

_x bn

+kβ k−1

×

_y

−x bn

y−x bn

+lβ l−1

1− y

bn

tk,ln (y, bn)

n−k−l−1 .

By the hypothesis, we can write

|G∗_n(f;y)−G∗_n(f;x)| ≤M(1 +nβ)1−n

n X

k=0

n−k X

l=0

l nbn

γ

n!

k!l! (n−k−l)!

x bn

x bn

+kβ k−1

_y−x bn

y−x bn

+lβ l−1

1− y

bn

tk,l_n (y, bn)

n−k−l−1

and by changing the order of the summations we have

|G∗_n(f;y)−G∗_n(f;x)| ≤M(1 +nβ)1−n

n X

l=0 _l

nbn γ_n

l

y−x bn

y−x bn

+lβ l−1

× n−l X

k=0 _n−l

k _x

bn _x

bn +kβ

k−1

1− y

bn

tk,l_n (y, bn)

n−k−l−1 .

From (1.3), we obtain

|G∗_n(f;y)−G∗_n(f;x)| ≤M(1 +nβ)1−n

n X

l=0 _l

nbn γ_n

l

y−x bn

y−x bn

+lβ l−1

×

1−(y−x)

bn

tl,_n0(y−x, bn) n−l−1

Considering the H¨older inequality withp= 1/γ andq= 1/(1−γ),we have |G∗_n(f;y)−G∗_n(f;x)|

≤M (

(1 +nβ)1−n n X

l=0 l nbn

_n l

y−x bn

y−x bn

+lβ l−1

×

1−(y−x)

bn

tl,_n0(y−x, bn) n−l−1

γ

×

(

(1 +nβ)1−n n X

l=0

n l

y−x bn

y−x bn

+lβ l−1

×

1−(y−x)

bn

tl,_n0(y−x, bn)

n−l−11−γ

=M{G∗_n(t;y−x)}γ.{G∗_n(1;y−x)}γ =M(y−x)γ.

From (1.7), (1.8) and by the hypothesis, we get

|G∗_n(f;y)−G_n∗(f;x)| ≤ |y−x|γ.

This proves the theorem.

Theorem 3. If f isconvex,then G∗_n(f;x)≥f(x). Proof. Suppose that

αk = (1 +nβ)1− nn

k

x bn

x bn

+kβ k−1

1− x

bn

tk,n0(x, bn) n−k−1

and

xk =

k

nbn, k= 0,1,2..., n.

It is easy to see that

n X

k=0

αk =G∗n(1;x) = 1. Therefore we obtain

G∗n(f;x) = n X

k=0

αkf(xk)≥f n X

k=0 αkxk

!

=f(G∗n(t;x)) =f(x),

which completes the proof.

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Dilek S¨oylemez

Ankara University, Elmadag Vocational School, Department of Computer Pogram-ming, 06780, Ankara, Turkey

E-mail address:[email protected]

Fatma Tas¸delen

Ankara University, Faculty of Science, Department of Mathematics, 06100, Tandogan, Ankara, Turkey