Order of approximation by an operator involving biorthogonal polynomials

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R E S E A R C H

Open Access

Order of approximation by an operator

involving biorthogonal polynomials

Özlem Öksüzer

1*

_{, Harun Karsli}

2

_{and Fatma Ta¸sdelen Ye¸sildal}

1

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Faculty of Science, University of Ankara, Ankara, Turkey Full list of author information is available at the end of the article

Abstract

The goal of this paper is to estimate the rate of convergence of a linear positive operator involving Konhauser polynomials to bounded variation functions on [0, 1]. To prove our main result, we have used some methods and techniques of probability theory.

MSC: 41A25; 41A35

Keywords: Konhauser polynomials; order of convergence; functions of bounded variation; Lebesgue-Stieltjes integration

1 Introduction

In , Konhauser presented the general theory of biorthogonal polynomials []. After-wards, in  [], he gave the following pair of biorthogonal polynomials:Yν(n)(x;k) and

Z(n)

ν (x;k) (n> –,k∈Z+), satisfying

Yν(n)(x; ) =Zν(n)(x; ) =L(n)ν (x),

whereL(n)

ν (x) are classical Laguerre polynomials and Yν(n)(x;k) Konhauser polynomials

given by

L(n)_v (x) = v

j= (–)j

v+n v–j

xj

j!,

and

Yν(n)(x;k) =

 ν!

ν

i=

xi

i! i

j= (–)j

i j

j+n+ 

k

ν

.

The classical Meyer-König and Zeller (MKZ) operators are deﬁned in  [] by

Mn(f;x) = ( –x)n+

∞

k=

f

k k+n+ 

n+k k

xk, x∈[, ),

Mn(f; ) =f().

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In order to give the monotonicity properties, Cheney and Sharma introduced the follow-ing type modiﬁcation of the MKZ operators:

M∗_n(f;x) = ( –x)n+

∞

k=

f

k k+n

n+k k

xk, x∈[, ), ()

M∗_n(f; ) =f().

In [], they also introduced the operators forx∈[, ) andt∈(–∞, ]

Pn(f;x) =exp

tx

 –x

∞

k=

f

k k+n

L(n)_k (t)xk( –x)n+, ()

Pn(f; ) =f(),

whereL(n)_k (t) denotes the classical Laguerre polynomials.

We consider the sequence of linear positive operators (similarly to []), which is an-other generalization of these operators includingYυ(n)(x;k) Konhauser polynomials. For

x∈[, ),t∈(–∞, ], andk<n+ 

(Lnf)(x,t;k) = 

Fn(x,t)

∞

υ=

f

υk k(υ– ) +n+ 

Y_υ(n)(t;k)xυ_, _()

where {Fn(x,t)}n∈N are the generating functions for the sequence of functions

{Y(n)

υ (x;k)}v∈N, namely

Fn(x,t) =

∞

υ=

Yυ(n)(t;k)x υ

, n> , ()

and

Fn(x,t) = ( –x)–nk+_exp_–_t_{( –}_x₎–k_{– } _. _()

Y(n)

υ (t;k)≥ fort∈(–∞, ], due to Carlitz []. If we choosek=  in (), then the operators

turn out to be (). Similarly, if we choosek=  andt=  in (), we get (), which are called Bernstein power series by Cheney and Sharma in [].

In view of () and (), one has

 = ( –x)(n+)/kexpt( –x)–/k– 

∞

υ=

Y_υ(n)(t;k)xυ_, _n_{> .}

For simplicity, we set

mn(x,t;k) = ( –x)(n+)/kexpt( –x)–/k–  Yυ(n)(t;k)x υ

. ()

In view of (), we can write () as

(Lnf)(x,t;k) =

∞

υ=

f

υk k(υ– ) +n+ 

mn(x,t;k), ()

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Our paper concerns the rate of pointwise convergence of the operators given by (). In particular, by means of the techniques of probability theory, we shall estimate the rate of convergence for the operators (Lnf) for functions of a bounded variation on [, ] at points

xwheref(x+) andf(x–) exist.

It is worthwhile to say that our present results extend some earlier results. In fact, if we choosek=  inLnf, then the operators reduce to the operators mentioned [], and ifk=  andt=  inLnf, then we get the operator investigated in [].

For some important papers on diﬀerent operators related to the present study we refer the readers to Bojanic [] and Bojanic and Vuilleumier [] where they estimated the rate of convergence of Fourier series and Fourier-Legendre series of functions of bounded vari-ation, respectively. Cheng [, ] gave two results of this type for the Bernstein operators and Szász operators. After the fundamental studies of Bojanic-Vuilleumier and Cheng, their methods have been used widely in many follow up investigations (see, for instance [, , –]).

The main theorem of this work reads as follows.

Theorem . Let f be a function of a bounded variation on[, ] (f ∈BV[, ]).Then for each x∈(, ),t∈(–∞, ],k<n+ ,and n suﬃciently large,we have

(Lnf)(x,t;k) –

f(x+) +f(x–) 

≤  n

(x– x+ )(k+ nx–t( –x)–/k(k+ x))

x( –x) + 

n

l=

x+(–x)/√l

x–x/√l (gx)

+ C

∗ √

nA(x,t;k)f(x+) –f(x–)+εn(x)f(x) –f(x–),

where

A(x,t;k)

=



ke

t–t(–x)–/k_{( –}_x₎–(+_kk)_x__et_{( –}_x₎_k _–_t

x

+ et+t(–x)–/kt– ( –x)k( –x)



kxk( –x)



k –t( –x)



k

+t– t( –x)k _{+ ( –}_x₎k_x

+et(–x)– k( –x)k__k_t_{– }_t_{( –}_x₎k _{+ ( –}_x₎k_{( –}_x₎k_x

+–t+ t( –x)k _{– }_t_{( –}_x₎k _{+ ( –}_x₎k_x

+k–t+ ( –x)k_{( –}_x₎k_{( +}_x₎

_

ke

t–t(–x)–/k

( –x)–(+kk)xe–tt– ( –x)



kx

+et(–x)– k( –x)k_k_{( –}_x₎k_–_t_{( –}_x₎k₊_t_{– }_t_{( –}_x₎k _{+ ( –}_x₎k_x

/ ,

εn(x) =

, x= υk

k(υ–)+n+ for some k∈ {, , , . . . ,n}, , x= υk

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C∗is a certain constant,b_a(gx)is a total variation of gxon[a,b]and

gx(s) = ⎧ ⎪ ⎨ ⎪ ⎩

f(s) –f(x+), x<s≤, , s=x,

f(s) –f(x–), ≤s<x.

2 Some lemmas

We now mention certain results which are necessary to prove our main theorem.

Lemma . For every x∈[, ),t∈(–∞, ],and k<n+ ,we have

(Ln)(x,t;k) = ,

(Lns)(x,t;k)≤x–

tx

n( –x)/k, ()

Lns

(x,t;k) <x– tx 

n( –x)/k –

ktx n( –x)/k+

kx n,

and

Ln(s–x)

(x,t;k)≤x(k+ nx–t( –x)

–/k₍_k_{+ }_x₎₎

n . ()

Proof For the proof of () see [].

To prove (), we can use the following equality:

Ln(s–x)(x,t;k) =

∞

v=

υk

k(υ– ) +n+ –x 

mn(x,t;k)

≤ ∞

v=

υk k(υ– ) +n+ 



mn(x,t;k)

+ x ∞

v=

υk k(υ– ) +n+ 

mn(x,t;k) +x ∞

v=

mn(x,t;k).

From (), one has (). So, this completes the proof of Lemma ..

Lemma . For x∈(, ),t≤,

λn(x,t,y;k) = y



Kn(x,t,s;k)ds, ≤s≤x

≤ x(k+ nx–t( –x)–/k(k+ x)) n(x–y) ,

where

Kn(x,t,y;k) =

k≤(n+)y v(–y)+y

mn(x,t;k),  <y< ,

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Proof Fors<y<x,

λn(x,t,y;k) = y



Kn(x,t,s;k)ds

≤

y



Kn(x,t,s;k)

x–s x–y



ds

=  (x–y)

y



Kn(x,t,s;k)(x–s)ds

≤ 

(x–y)

Ln(s–x)

(x,t;k)

≤ x(k+ nx–t( –x)–/k(k+ x))

n(x–y) .

Lemma . For x∈[, ),t∈(–∞, ],and k<n+ ,

∞

υ=

vYυ()(t;k)x υ

=e

t–t(–x)–/k_(–_t_{+ ( –}_x₎k_{)( –}_x₎––k_x

k , ()

∞

υ=

vYυ()(t;k)x υ₌ 

ke

t–t(–x)–/k_{( –}_x₎––_k

xk( –x)k –_t( –_x)



k

+t– t( –x)k _{+ ( –}_x₎k_x_, _()

∞

υ=

vYυ()(t;k)x υ

= 

ke

t–t(–x)–/k_{( –}_x₎––_k_x__k_t_{– }_t_{( –}_x₎_k_{( –}_x₎_k_{( –}_x₎_k_x

+–t+ t( –x)k_{– }_t_{( –}_x₎k _{+ ( –}_x₎k_x

+k–t+ ( –x)k_{( –}_x₎



k_{( +}_x₎_. _()

Proof Forn= , letYυ(n)(x;k) be the Konhauser polynomials’ generating function deﬁned

by

∞

υ=

Y_υ()(t;k)xυ_{= ( –}_x₎–_k_exp_–_t_{( –}_x₎–_k _{– } _.

Taking derivatives of both sides with respect toxyields

∞

υ=

vYυ()(t;k)x

υ–₌et–t(–x)

–/k

(–t+ ( –x)k_{)( –}_x₎––



k

k .

Editing the equation, we have

∞

υ=

vY_υ()(t;k)xυ₌et–t(–x)

–/k

(–t+ ( –x)k_{)( –}_x₎––k_x

k .

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Lemma . Letζbe the random variable with

P(ζ=k) = ( –x)/kexp

t( –x)–/k–  Yυ()(t;k)x υ

.

Then letting a=E|ζ|,b=E(ζ–a),

E(ζ–a)= 

ke

t–t(–x)–/k

( –x)–(+kk)_x_e–t_t_{– ( –}_x₎k_x

+et(–x)– k( –x)k_k( –_x)



k –_t( –_x)



k

+t– t( –x)k+ ( –x)



kx

and

E|ζ–a|= 

ke

t–t(–x)–/k_{( –}_x₎–(+_kk)_x__et_{( –}_x₎_k_–_t_x

+ et+t(–x)–/kt– ( –x)k_{( –}_x₎



k_x_k_{( –}_x₎



k_–_t_{( –}_x₎



k

+t_{– }_t_{( –}_x₎_k_{+ ( –}_x₎_k_x

+et(–x)– k( –x)k__k_t_{– }_t_{( –}_x₎k_{+ ( –}_x₎k_{( –}_x₎k_x

+–t+ t( –x)k_{– }_t_{( –}_x₎k _{+ ( –}_x₎k_x

+k–t+ ( –x)k_{( –}_x₎k_{( +}_x₎_.

Proof Forn= , we have

a=E(ζ) =

∞

υ=

vY_υ()(t;k)xυ₌et–t(–x)

–/k

(–t+ ( –x)k_{)( –}_x₎––k_x

k ,

b_=E(ζ–a)=E

ζ_– aE(ζ) +a=E

ζ_–E(ζ),

and

E|ζ–a|=E

ζ_– aE

ζ_+ a_E(ζ) –a_=Eζ_– E(ζ)Eζ_+ E(ζ).

If we use ()-(), the proof of the Lemma . is completed.

Next, we recall the well-known Berry-Esséen bound for the classical central limit theo-rem of probability theory.

Lemma .(Berry-Esséen) Let{ζk}∞k=be a sequence of independent and identically

dis-tributed random variables with ﬁnite variance such that the expectation E(ζ) =a∈R,

the varianceVar(ζ) =E(ζ–a)=b> .Assume E|ζ–a|<∞,then there exists a

con-stant C, /√π≤C≤.,such that for all n= , , . . .and all t,

P



b√n n

k=

(ζk–a)≤t

–√ π

t

–∞e

–u/_du <C

E|ζ–a|

b_√n .

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Lemma . For x∈(, ),t∈(–∞, ]and k<n+ ,we have

(n+)x v(–x)+x<k

mn(x,t;k) –  ≤√C

nA(x,t;k);

A(x,t;k)is given in Theorem..

Proof By direct calculation, one has from Lemmas .-. the desired result.

Lemma . For all x∈(, )deﬁne the functionsgn(s–x)by

sgn(s–x) = ⎧ ⎪ ⎨ ⎪ ⎩

, s>x, , s=x, –, s<x.

We have

Lnsgn(s–x)

(x,t;k) = 

(n+)x v(–x)+x<k

mn(x,t;k) –  +εn(x)mn

x,t;k,

whereεn(x)is as in Theorem.and k=_{k(v–)+n+}vk .

Proof If we apply the operatorLnto the function ofsgn(s–x), we have

Lnsgn(s–x)

(x,t) =

∞

v= sgn

υk

k(υ– ) +n+ –x

mn(x,t;k)

=

(n+)x v(–x)+x<k

mn(x,t;k) +

k<_v(_(–n+)_x₎₊x_x

(–)mn(x,t;k),

and we can write

(Ln)(x,t) =

(n+)x v(–x)+x<k

mn(x,t;k) +

k<_v(_(–n+)_x₎₊x_x

mn(x,t;k) +εn(x)mn

x,t;k,

k<_v(_(–n+)_x₎₊x_x

mn(x,t;k) =  –

(n+)x v(–x)+x<k

mn(x,t;k) –εn(x)mn

x,t;k.

Thus

Lnsgn(s–x)

(x,t;k)

=

(n+)x v(–x)+x<k

mn(x,t;k) –

 –

(n+)x v(–x)+x<k

mn(x,t;k) –εn(x)mn

x,t;k

= 

(n+)x v(–x)+x<k

mn(x,t;k) –  +εn(x)mn

x,t;k.

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Lemma . If the conditions of Theorem.hold,we have for all x∈(, )

f(x+) –f(x–) 

Lnsgn(s–x)

(x,t;k) +

f(x) –f(x+) +f(x–) 

(Lnδ)(x,t;k)

≤√C∗

nA(x,t;k)f(x+) –f(x–)+εn(x)f(x) –f(x–), ()

whereδx(s)is the Dirac delta function,C∗ is a certain constant,and A(x,t;k)is given in Theorem..

Proof By direct calculation, we get

(Lnδ)(x,t;k) =εn(x)mn

x,t;k.

One has

f(x+) –_f(x–)Lnsgn(s–x)

(x,t;k) +

f(x) –f(x+) +f(x–) 

(Lnδ)(x,t;k)

≤f(x+) –f(x–)



Lnsgn(s–x)

(x,t;k)+

f(x) –f(x+) +f(x–) 

(Lnδ)(x,t;k)

≤f(x+) –f(x–)



(n+)x v(–x)+x<k

mn(x,t;k) –  +εn(x)mn

x,t;k

+

f(x) –f(x+) +f(x–) 

εn(x)mn

x,t;k

≤f(x+) –f(x–)

(n+)x v(–x)+x<k

mn(x,t;k) – 

+f(x) –f(x–)εn(x)mn

x,t;k.

According to Lemma ., one has

(n+)x v(–x)+x<k

mn(x,t;k) –  ≤√C

nA(x,t;k), ()

and using the method of proof of Lemma . and evaluations which are similar to the work in [], we have

mn

x,t;k=:√C

nA(x,t;k). ()

SetC∗=max{C,C}. Consequently from () and () we get ().

This completes the proof of Lemma ..

Lemma . For n suﬃciently large,we have

(Lngx)(x,t;k)

≤  n

(x_{– }_x_{+ )(}_k_{+ }_nx_–_t_{( –}_x₎–/k₍_k_{+ }_x₎₎

x( –x) + 

n

l=

x+(–x)/√l

x–x/√l (gx)

,

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Proof We recall the Lebesgue-Stieltjes integral representations

(Lnf)(x,t;k) = 



f(y)dyλn(x,t,y;k), ()

whereλn(x,t,y;k) = y

Kn(x,t,s;k)ds, ≤s≤x. From (), we can rewrite (Lngx)(x,t;k) as follows:

(Lngx)(x,t;k)=

∞

v=

gx

υk k(υ– ) +n+ 

mn(x,t;k)

= 



gx(y)dyλn(x,t,y;k)

. ()

To estimate (), we decompose it into three parts as follows:





gx(y)dyλn(x,t,y;k) = (Igx)(x,t;k) + (Igx)(x,t;k) + (Igx)(x,t;k), ()

where

(Igx)(x,t) =

x–x/√n



gx(y)dyλn(x,t,y;k),

(Igx)(x,t) =

x+(–x)/√n

x–x/√n

gx(y)dyλn(x,t,y;k),

(Igx)(x,t) = 

x+(–x)/√n

gx(y)dyλn(x,t,y;k).

We shall evaluate (Igx)(x,t), (Igx)(x,t), and (Igx)(x,t).

First we estimate (Igx)(x,t), fory∈[x–x/√n,x+ ( –x)/√n]. Note thatgx(x) =  and|

x+(–x)/√n

x–x/√n dyλn(x,t,y;k)| ≤, so we have

(Igx)(x,t;k)=

x+(–x)/

√

n

x–x/√n

gx(y) –gx(x)dyλn(x,t,y;k)

≤

x+(–x)/√n

x–x/√n

gx(y) –gx(x)dyλn(x,t,y;k)

≤

x+(–x)/√n

x–x/√n

(gx)≤ 

n–  n

l=

x+(–x)/√l

x–x/√l

(gx). ()

Next we estimate (Igx)(x,t). Using partial Lebesgue-Stieltjes integrations, we obtain

(Igx)(x,t;k) =

x–x/√n



gx(y)dyλn(x,t,y;k)

=gx(x–x/√n)λn(x,t,x–x/√n;k) –gx()λn(x,t, ;k)

–

x–x/√n



(10)

Since|gx(x–x/√n)|=|gx(x–x/√n) –gx(x)| ≤x_x–x/√_n(gx), it follows that

(Igx)(x,t;k)≤ x

x–x/√n

(gx)λn(x,t,x–x/

√ n;k) +

x–x/√n



λn(x,t,y;k)dy

– x

y (gx)

.

From Lemma ., it is clear that

λn(x,t,x–x/√n;k)≤x(k+ nx–t( –x)

–/k₍_k_{+ }_x₎₎

n(√x

n)

.

It follows that

(Igx)(x,t;k)≤ x

x–x/√n

(gx)x(k+ nx–t( –x)

–/k₍_k_{+ }_x₎₎

n(_√x n)

+

x–x/√n



x(k+ nx–t( –x)–/k₍_k_{+ }_x₎₎

n(x–y) dy

– x

y (gx)

= x

x–x/√n (gx)

x(k+ nx–t( –x)–/k₍_k_{+ }_x₎₎

n(√x

n)

+x(k+ nx–t( –x)

–/k₍_k_{+ }_x₎₎

n

x–x/√n



 (x–y)dy

–

x

y (gx)

.

Furthermore, since

x–x/√n



 (x–y)dy

–

x

y (gx)

= –  (x–y)

x

y

(gx)x–x/

√

n

 +

x–x/√n



 (x–y)

x

y (gx)dy

= –  (√x

n) 

x

x–x/√n (gx) + 

x x

 (gx) +

x–x/√n



 (x–y)

x

y (gx)dy,

and puttingy=x–x/√uin the last integral, we get

x–x/√n



 (x–y)

x

y

(gx)dy=  x n  x

x–x/√u

(gx)du=  x n l= x

x–x/√l (gx).

Consequently,

(Igx)(x,t)≤ x

x–x/√n

(gx)x(k+ nx–t( –x)

–/k₍_k_{+ }_x₎₎

n(√x

n)

+x(k+ nx–t( –x)

–/k₍_k_{+ }_x₎₎

n

– 

(√x

n) 

x

(11)

+ 

x x



(gx) + 

x n

l= x

x–x/√l (gx)

= x(k+ nx–t( –x)

–/k₍_k_{+ }_x₎₎

n



x x



(gx) + 

x n

l= x

x–x/√l (gx)

= k+ nx–t( –x)

–/k₍_k_{+ }_x₎

nx

_x

 (gx) +

n

l= x

x–x/√l (gx)

.

Moreover,

x

 (gx)≤

n

l= x

x–x/√l (gx)≤

n

l=

x+(–x)/√l

x–x/√l (gx)

yields

(Igx)(x,t)≤

(k+ nx–t( –x)–/k₍_k_{+ }_x₎₎

nx

_n

l=

x+(–x)/√l

x–x/√l (gx)

. ()

Using a similar method for estimating|(Igx)(x,t)|, we get

(Igx)(x,t)≤

x(k+ nx–t( –x)–/k(k+ x))

n( –x)

_n

l=

x+(–x)/√l

x (gx)

. ()

Putting ()-() in (), we get

(Lngx)(x,t)≤(Igx)(x,t)+(Igx)(x,t)+(Igx)(x,t)

≤ (k+ nx–t( –x)–/k(k+ x)) nx

_n

l=

x+(–x)/√l

x–x/√l (gx)

+ 

n–  n

l=

x+(–x)/√l

x–x/√l (gx)

+x(k+ nx–t( –x)

–/k₍_k_{+ }_x₎₎

n( –x)

_n

l=

x+(–x)/√l

x (gx)

. ()

Obviously,



x+ x

( –x) =

x_{– }_x_{+ }

x( –x)

and

x+(–x)/√l

x

(gx)≤

x+(–x)/√l

(12)

Hence, we obtain from ()

(Lngx)(x,t)≤



x+ x

( –x)

k+ nx–t( –x)–/k₍_k_{+ }_x₎

n

l=

x+(–x)/√l

x–x/√l (gx)

+ 

n–  n

l=

x+(–x)/√l

x–x/√l (gx)

≤  n

(x_{– }_x_{+ )(}_k_{+ }_nx_–_t_{( –}_x₎–/k₍_k_{+ }_x₎₎

x( –x) + 

×

_n

l=

x+(–x)/√l

x–x/√l (gx)

.

This inequality is equivalent to the one to be proved.

3 Proof of the main theorem

Now we are ready to establish our main theorem.

Proof For anyf ∈BV[, ], we decomposefinto four parts on [, ] for suﬃciently largen,

f(s) =f(x+) +f(x–)

 +gx(s) +

f(x+) –f(x–)

 sgn(s–x)

+δx(s)

f(x) –f(x+) +f(x–) 

, ()

where

δx(s) =

, x=s, , x=s.

If we apply the operatorLnto the both sides of (), then we have

(Lnf)(x,t;k) =

f(x+) +f(x–)

 (Ln)(x,t;k) + (Lngx)(x,t;k)

+f(x+) –f(x–) 

Lnsgn(s–x)

(x,t;k)

+

f(x) –f(x+) +f(x–) 

(Lnδx)(x,t;k).

It follows that

(Lnf)(x,t;k) –

f(x+) +f(x–) 

≤(Lngx)(x,t;k)+

f(x+) –_f(x–)Lnsgn(s–x)

(x,t;k)

+

f(x) – f(x+) +f(x–) 

(Lnδx)(x,t;k) .

(13)

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Faculty of Science, University of Ankara, Ankara, Turkey.}2_{Department of Mathematics,}

Faculty of Science and Arts, Abant Izzet Baysal University, Bolu, Turkey.

Received: 23 October 2014 Accepted: 30 March 2015

References

1. Konhauser, JDE: Some properties of biorthogonal polynomials. J. Math. Anal. Appl.11(1-3), 242-260 (1965) 2. Konhauser, JDE: Biorthogonal polynomials suggested by the Laguerre polynomials. Pac. J. Math.21, 303-314 (1967) 3. Meyer-König, W, Zeller, K: Bernsteinsche potenzreihen. Stud. Math.19, 89-94 (1960)

4. Cheney, EW, Sharma, A: Bernstein power series. Can. J. Math.16, 241-252 (1964)

5. Do ˘gru, O, Özarslan, MA, Ta¸sdelen, F: On positive operators involving a certain class of generating functions. Studia Sci. Math. Hung.41(4), 415-429 (2004)

6. Carlitz, L: A note on certain biorthogonal polynomials. Pac. J. Math.24, 425-430 (1968)

7. Khan, MK: On the rate of converge of Bernstein power series for functions of bounded variation. J. Approx. Theory57, 90-103 (1989)

8. Zeng, XM: Rates of approximation of bounded variation functions by two generalized Meyer-König and Zeller type operators. Comput. Math. Appl.39, 1-13 (2000)

9. Bojanic, R: An estimate of the rate of convergence for Fourier series of functions of bounded variation. Publ. Inst. Math. (Belgr.)26(40), 57-60 (1979)

10. Bojanic, R, Vuilleumier, M: On the rate of convergence of Fourier-Legendre series of functions of bounden variation. J. Approx. Theory31, 67-79 (1981)

11. Cheng, F: On the rate of convergence of Bernstein polynomials of functions of bounded variation. J. Approx. Theory 39, 259-274 (1983)

12. Cheng, F: On the rate of convergence of Szasz-Mirakyan operator of functions of bounded variation. J. Approx. Theory40, 226-241 (1983)

13. Guo, S: On the rate of convergence of the integrated Meyer-König and Zeller operators for functions of bounded variation. J. Approx. Theory56, 245-255 (1989)

14. Pych-Taberska, P, Karsli, H: On the rates of convergence of Bernstein-Chlodovsky polynomials and their Bézier-type variants. Appl. Anal.90(3-4), 403-416 (2011)

15. Öksüzer, Ö, Karsli, H, Ta¸sdelen, F: Convergence rate of Bézier variant of an operator involving Laguerre polynomials of degreen. In: 11th International Conference of Numerical Analysis and Applied Mathematics 2013: ICNAAM 2013. Aip Conference Proceeding, vol. 1558, 1160 (2013). doi:10.1063/1.4825714

16. ˙Içöz, G, Ta¸sdelen, F, Varma, S: On linear positive operators involving biorthogonal polynomial. Ars Comb.105, 319-331 (2012)