On approximation by some Bernstein–Kantorovich exponential-type polynomials

6 years ago

(original), Approximation Theory, ISI/JCR, paper

Abstract

Since the introduction of Bernstein operators, many authors defined and/or studied Bernstein type operators and their generalizations, among them are Morigi and Neamtu (Adv Comput Math 12:133–149, 2000). They proposed an analog of classical Bernstein operator and proved some convergence results for continuous functions.

Herein, we introduce their integral extensions in Kantorovich sense by replacing the usual differential and integral operators with their more general analogues. By means of these operators, we are able to reconstruct the functions which are not necessarily continuous. It is shown that the operators form an approximation process in both $C [0, 1]$ and $L_{p, μ} [0, 1]$ , which is an exponentially weighted space.

Also, quantitative results are stated in terms of appropriate moduli of smoothness and K-functionals. Furthermore, a quantitative Voronovskaya type result is presented.

Authors

Ali Aral

Diana Otrocol
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Ioan Raşa

Keywords

Bernstein–Kantorovich operator; uniform convergence; modulus of continuity.

References

See the expanding block below.

Paper coordinates

A. Aral, D. Otrocol, I. Raşa, On approximation by some Bernstein–Kantorovich exponential-type polynomials, Periodica Mathematica Hungarica, 79 (2019) 2, pp. 236-254,
doi: 10.1007/s10998-019-00284-3

PDF

not available yet.

About this paper

Journal

Periodica Mathematica Hungarica

Publisher Name

Springer

DOI

Print ISSN

0031-5303

Online ISSN

1588-2829

Google Scholar Profile

not yet

[1] T. Acar, A. Aral, S.A. Mohiuddine, Approximation by bivariate(p; q)-Bernstein-Kantorovich operators. Iran. J. Sci. Technol. Trans. A Sci. 42(2), 655–662 (2018)MathSciNetCrossRefGoogle Scholar

[2] F. Altomare, M. Campiti, Korovkin-type Approximation Theory and Its Applications (Walter de Gruyter, Berlin, 1994)CrossRefGoogle Scholar

[3] F. Altomare, V. Leonessa, On a sequence of positive linear operators associated with a continuous selection of Borel measures. Mediterr. J. Math. 3, 363–382 (2006)MathSciNetCrossRefGoogle Scholar

[4] F. Altomare, M. Cappelletti Montano, V. Leonessa, I. Raşa, A generalization of Kantorovich operators for convex compact subsets. Banach J. Math. Anal. 11(3), 591–614 (2017)MathSciNetCrossRefGoogle Scholar

[5] F. Altomare, M. Cappelletti Montano, V. Leonessa, I. Raşa, Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators. J. Math. Anal. Appl. 458(1), 153–173 (2018)MathSciNetCrossRefGoogle Scholar

[6] A. Aral, V. Gupta, R.P. Agarwal, Applications of q-Calculus in Operator Theory (Springer, New York, 2013)CrossRefGoogle Scholar

[7] A. Aral, D. Cárdenas-Morales, P. Garrancho, Bernstein-type operators that reproduce exponential functions. J. Math. Inequal. (Accepted)Google Scholar

[8] R.A. Devore, “Degree of approximation,” in ApproximationTheory II (Academic Press, New York, 1976), pp. 117–161Google Scholar

[9] Z. Ditzian, X. Zhou, Kantorovich–Bernstein polynomials. Constr. Approx. 6, 421–435 (1990)MathSciNetCrossRefGoogle Scholar

[10] H. Gonska, P. Piţul, I. Raşa, On Peano’s form of the Taylor remainder, Voronovskaja’s theorem and the commutator of positive linear operators, in Proceedings of the International Conference on Numerical Analysis and Approximation Theory (Cluj Napoca, Romania), July 5–8, pp. 55–80 (2006)Google Scholar

[11] K.G. Ivanov, Approximation by Bernstein polynomials in

L_{p}

[12] L.V. Kantorovich, Sur certains d’eveloppements suivant les polynmes de la forme de S. Bernstein, I, II. C. R. Acad. URSS, pp. 563–568 and 595–600 (1930)Google Scholar

[13] G.G. Lorentz, Approximation of Functions (Chelsea Publ. Co., New York, 1986)zbMATHGoogle Scholar

[14] S. Morigi, M. Neamtu, Some results for a class of generalized polynomials. Adv. Comput. Math. 12, 133–149 (2000)MathSciNetCrossRefGoogle Scholar

[15] J. Peetre, A theory of interpolation of normed spaces, Lecture Notes, Brasilia, 1963 (Notas de Matematica, 39 (1968)Google Scholar

[16] R. Păltănea, A note on Bernstein–Kantorovich operators. Bull. Univ. Transilvania of Braşov, Series III 6(55)(2), 27–32 (2013)zbMATHGoogle Scholar

[17] P.C. Sikkema, Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen. Numer. Math. 3, 107–116 (1961)MathSciNetCrossRefGoogle Scholar

[18] E.M. Stein, Singular Integrals (Princeton, New Jersey, 1970)zbMATHGoogle Scholar

[19] A. Zygmund, Trigonometric Series I, II (Cambridge University Press, Cambridge, 1959)zbMATHGoogle Scholar

∎

¹¹institutetext: Ali Aral ²²institutetext: Kırıkkale University, Faculty of Sciences and Arts, Department of Mathematics, 71450, Kırıkkale-Turkey

²²email: aliaral73@yahoo.com ³³institutetext: Diana Otrocol ⁴⁴institutetext: Technical University of Cluj-Napoca, Faculty of Automation and Computer Science, Department of Mathematics, Memorandumului St. no. 28, Cluj-Napoca 400114, Romania; Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, P.O.Box. 68-1, 400110, Cluj-Napoca, Romania

⁴⁴email: Diana.Otrocol@math.utcluj.ro ⁵⁵institutetext: Ioan Raşa ⁶⁶institutetext: Technical University of Cluj-Napoca, Faculty of Automation and Computer Science, Department of Mathematics, Memorandumului St. no. 28, Cluj-Napoca 400114, Romania

⁶⁶email: ioan.rasa@math.utcluj.ro

On approximation by some Bernstein-Kantorovich exponential-type polynomials

Ali Aral Diana Otrocol and Ioan Raşa

(Received: date / Accepted: date)

Abstract

Since the introduction of Bernstein operators, many authors defined and/or studied Bernstein type operators and their generalizations, among them are Morigi and Neamtu in 2000. They proposed an analog of classical Bernstein operator and proved some convergence results for continuous functions. Herein, we introduce their integral extensions in Kantorovich sense by replacing the usual differential and integral operators with their more general analogs. By means of these operators, we are able to reconstruct the functions which are not necessarily continuous. It is shown that the operators form an approximation process in both $C\left[0,1\right]$ and $L_{p,\mu}\left[0,1\right]$ , which is an exponentially weighted space. Also, quantitative results are stated in terms of appropriate moduli of smoothness and $K$ -functionals. Furthermore, a quantitative Voronovskaya type result is presented.

1 Introduction

The classical Bernstein operators associate the polynomial $B_{n}f$ to any function $f\in C[0,1]$ and the association is defined by

B_{n}\left(f\right)\left(x\right)=\sum\limits_{k=0}^{n}f\left(\frac{k}{n}\right)p_{n,k}\left(x\right),\ n\in\mathbb{N},

(1)

where

p_{n,k}(x)={n\choose k}x^{k}(1-x)^{n-k}.

In fact, it defines a linear approximation process in the space of all continuous functions on $\left[0,1\right].$ To obtain an approximation process for Lebesgue integrable functions, the Kantorovich version of (1) was defined in Kantorovich by replacing the sample values $f(k/n)$ with the mean values of $f$ in the interval $\left[\frac{k}{n},\frac{k+1}{n}\right]$ , that is

K_{n}\left(f\right)\left(x\right)=\left(n+1\right)\sum\limits_{k=0}^{n}p_{n,k}\left(x\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}f\left(t\right)dt,x\in\left[0,1\right],\ n\in\mathbb{N},

where $f:\left[0,1\right]\rightarrow\mathbb{R}$ is a locally integrable function. We note that $B_{n}$ and $K_{n}$ are connected by the relation

K_{n}=D\circ B_{n+1}\circ I,

(2)

where $D$ is the differentiation operator (i.e. $D\left(f\right)=f^{{}^{\prime}}$ , $f\in C^{1}\left[0,1\right]$ ) and $I$ is the antiderivative operator (i.e. $I\left(f;x\right)=\int_{0}^{x}f\left(t\right)dt,$ $f\in C\left[0,1\right]$ ) and $x\in\left[0,1\right]$ . These operators allow us to reconstruct a Lebesgue integrable function by means of its mean values on the sets $\left[\frac{k}{n},\frac{k+1}{n}\right]$ . It is well known that Bernstein operators reproduce $e_{i}\left(x\right)=x^{i},$ $i=0,1$ whereas Bernstein-Kantorovich operators reproduce only $e_{0}\left(x\right)=1.$

Recently, Aral et al. Ar-Card-Gar studied a sequence of linear positive operators which generalize the classical Bernstein operators and perform better than $B_{n}$ under sufficient conditions. These operators reproduce the exponential functions $\exp(\mu t)$ and $\exp(2\mu t)$ , $\mu>0,$ and are defined by

G_{n}f\left(x\right)=G_{n}\left(f;x\right)=\sum\limits_{k=0}^{n}f\left(\frac{k}{n}\right)e^{-\mu k/n}e^{\mu x}p_{n,k}\left(a_{n}\left(x\right)\right),x\in\left[0,1\right],n\in\mathbb{N},

where

a_{n}\left(x\right)=\frac{e^{\mu x/n}-1}{e^{\mu/n}-1}.

(3)

They have close connection with the Bernstein operators that is given by

G_{n}f\left(x\right)=\exp_{\mu}\left(x\right)B_{n}\left(\frac{f}{\exp_{\mu}};a_{n}\left(x\right)\right),

(4)

where for a fixed real parameter $\mu>0$ , the exponential function is defined as $\exp_{\mu}\left(x\right)=e^{\mu x}$ . We note that the generalization is a special case of the modification introduced by Morigi and Neamtu in Morigi-Nematu .

In order to give the generalization of the operators $K_{n}\left(f\right)$ , in Radu2 Păltănea considered the operators $D_{\mu}:C^{1}\left[0,1\right]\rightarrow C\left[0,1\right]$ and $I_{\mu}:C\left[0,1\right]\rightarrow C^{1}\left[0,1\right]$ , and defined the operators $K_{n}^{\mu}$ as

K_{n}^{\mu}=D_{\mu}\circ B_{n+1}\circ I_{\mu},

(5)

where

I_{\mu}\left(f,x\right)=e^{\mu x}\int_{0}^{x}e^{-\mu t}f\left(t\right)dt,f\in C\left[0,1\right]\mbox{and}\ x\in\left[0,1\right],

(6)

D_{\mu}\left(f,x\right)=f^{{}^{\prime}}\left(x\right)-\mu f\left(x\right),f\in C^{1}\left[0,1\right]\mbox{and}\ x\in\left[0,1\right].

(7)

Using the technique given in Radu2 with the operators in (5), we aim to construct a generalization of the operator $G_{n}$ and call it as generalized Bernstein Kantorovich operator. The construction is described in Section 2. Moreover, certain elementary properties including exponential moments are given in the same section. As we will see in Lemma 2 below, these operators reproduce only the function $\exp\left(2\mu t\right)$ , $\mu>0.$ In the same section, we also show that the new operator can be defined with the help of classical Bernstein operator and Bernstein-Kantorovich operator. In Section 3, it is shown that these operators form an approximation process in $C\left[0,1\right]$ by obtaining uniform convergence of them and an estimate in terms of modulus of continuity. In Section 4, a quantitative Voronovskaya type result is presented. In Section 5, we show that the new operators converge to the function in $L_{p,\mu}$ -norm. Furthermore, the approximation error of the operators is expressed in terms of an appropriate integral modulus of continuity.

The obtained results show that our technique is more convenient in order to introduce a Kantorovich version of $G_{n}.$

We mention that Kantorovich type operators have been object of the investigation by several mathematicians. Altomare et al. in Altomare-Va introduced a unitary approach to the study of the approximation properties for a large class of Kantorovich type operators including the classical Bernstein-Kantorovich operators and their various generalizations. Also, more results related to the mentioned operators can be found in Altomare2 and Altomare 3 . Various results on Kantorovich-type operators were given in Zhou and Ivanov . It is worth mentioning that Kantorovich type operators in classical sense were considered in the aforementioned more general framework, as well as in the context of the $q$ -operators and $\left(p,q\right)$ -operators, a very active area of research (see Aral-Gupta , tuncer- aral and the references therein).

2 Preliminaries

Throughout this section, we introduce a new family of operators and present some elementary properties.

Definition 1

Let $\mu>0$ . For any $n\in\mathbb{N}$ and $x\in\left[0,1\right]$ consider the operator $\widetilde{K}_{n}:C\left[0,1\right]\rightarrow C\left[0,1\right]$ , defined by

\widetilde{K}_{n}f\left(x\right)=a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu t}f\left(t\right)dt,

(8)

where $a_{n+1}\left(x\right)$ is given in (3).

To represent the operators in (8), we sometimes use the notation $\widetilde{K}_{n}\left(f;x\right)$ .

For a given $f\in L_{1}\left[0,1\right],$ we define $F_{\mu}\in C\left[0,1\right]$ as

F_{\mu}\left(x\right)=\int_{0}^{x}e^{-\mu t}f\left(t\right)dt.

(9)

Then, we have

\widetilde{K}_{n}f\left(x\right)=a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\left[F_{\mu}\left(\frac{k+1}{n+1}\right)-F_{\mu}\left(\frac{k}{n+1}\right)\right].

Using the notation

\delta_{n}^{\mu}\left(F\right)\left(x\right):=F_{\mu}\left(\frac{n}{n+1}x+\frac{1}{n+1}\right)-F_{\mu}\left(\frac{n}{n+1}x\right)

(10)

the operators $\widetilde{K}_{n}$ can be represented as

$\displaystyle\widetilde{K}_{n}f\left(x\right)$	$\displaystyle=a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)\exp_{\mu}\left(x\right)\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\delta_{n}^{\mu}\left(F\right)\left(\frac{k}{n}\right)$
	$\displaystyle=a_{n+1}^{{}^{\prime}}\left(x\right)\exp_{\mu}\left(x\right)K_{n}\left(\exp_{\mu}^{-1}\left(\cdot\right)f\left(\cdot\right);a_{n+1}\left(x\right)\right)$
	$\displaystyle=a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)\exp_{\mu}\left(x\right)B_{n}\left(\delta_{n}^{\mu}\left(F\right);a_{n+1}\left(x\right)\right),$	(11)

where $B_{n}$ and $K_{n}$ denote Bernstein and Bernstein-Kantorovich operators, respectively. Note that $\widetilde{K}_{n}f(x)$ is an exponential polynomial, based on the representation of Bernstein polynomials.

When we compare with the definition of the operators in Radu2 , the above relationships show that the approach which will be used below is more convenient to apply to the generalized Bernstein operators $G_{n},$ which preserve exponential functions. As we can see in Radu2 , the operator $K_{n}^{\mu}$ preserves good properties of Kantorovich operator $K_{n}$ . So we can expect that $\widetilde{K}_{n}$ also can preserve good properties of $K_{n}.$

Our process depends on the following lemma from Radu2 .

Lemma 1

Let $n\in\mathbb{N}$ and $x\in\left[0,1\right]$ . Then

a)

$\left(D_{\mu}\circ I_{\mu}\right)\left(f\right)\left(x\right)=f\left(x\right)$ , for all $f\in C\left[0,1\right]$ ,
b)

$\left(I_{\mu}\circ D_{\mu}\right)\left(f\right)\left(x\right)=f\left(x\right)$ , for all $f\in C^{1}\left[0,1\right]$ , with $f\left(0\right)=0$ .

Note that $D_{\mu}$ and $I_{\mu}$ in Lemma 1 are defined as in (6) and (7), respectively.

Theorem 2.1

Let $n\in\mathbb{N}$ and $x\in\left[0,1\right]$ . Then

\widetilde{K}_{n}=D_{\mu}\circ G_{n+1}\circ I_{\mu}.

Proof

Let $f\in C\left[0,1\right]$ and $x\in\left[0,1\right].$ With the help of the relations in Lemma 1 and the representation given in (4), we obtain

	$\displaystyle\left(D_{\mu}\circ G_{n+1}\circ I_{\mu}\right)\left(f\right)\left(x\right)$	$\displaystyle=\left(G_{n+1}\left(I_{\mu}\left(f\right);x\right)\right)^{{}^{\prime}}-\mu G_{n+1}\left(I_{\mu}\left(f\right);x\right)$
		$\displaystyle\hskip-56.9055pt=\left(\exp_{\mu}\left(x\right)B_{n+1}\left(F_{\mu};a_{n+1}\left(x\right)\right)\right)^{{}^{\prime}}-\mu\exp_{\mu}\left(x\right)B_{n+1}\left(F_{\mu};a_{n+1}\left(x\right)\right)$
		$\displaystyle\hskip-56.9055pt=\mu\exp_{\mu}\left(x\right)B_{n+1}\left(F_{\mu};a_{n+1}\left(x\right)\right)+\exp_{\mu}\left(x\right)B_{n+1}^{{}^{\prime}}\left(F_{\mu};a_{n+1}\left(x\right)\right)$
		$\displaystyle-\mu\exp_{\mu}\left(x\right)B_{n+1}\left(F_{\mu};a_{n+1}\left(x\right)\right),$

where $F_{\mu}$ defined as in (9). Using the fact that

\left(p_{n+1,k}\left(a_{n+1}\left(x\right)\right)\right)^{{}^{\prime}}=\left(n+1\right)a_{n+1}^{{}^{\prime}}\left(x\right)\left(p_{n,k-1}\left(a_{n+1}\left(x\right)\right)-p_{n,k}\left(a_{n+1}\left(x\right)\right)\right)

we have

	$\displaystyle\left(D_{\mu}\circ G_{n+1}\circ I_{\mu}\right)\left(x\right)$
		$\displaystyle\hskip-71.13188pt=e^{\mu x}\sum\limits_{k=0}^{n+1}\left(p_{n+1,k}\left(a_{n+1}\left(x\right)\right)\right)^{{}^{\prime}}F_{\mu}\left(\frac{k}{n+1}\right)$
		$\displaystyle\hskip-71.13188pt=\left(n+1\right)a_{n+1}^{{}^{\prime}}\left(x\right)e^{\mu x}\sum\limits_{k=0}^{n+1}\left(p_{n,k-1}\left(a_{n+1}\left(x\right)\right)-p_{n,k}\left(a_{n+1}\left(x\right)\right)\right)F_{\mu}\left(\frac{k}{n+1}\right)$
		$\displaystyle\hskip-71.13188pt=\left(n+1\right)a_{n+1}^{{}^{\prime}}\left(x\right)e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\left[F_{\mu}\left(\frac{k+1}{n+1}\right)-F_{\mu}\left(\frac{k}{n+1}\right)\right]$
		$\displaystyle\hskip-71.13188pt=a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu t}f\left(t\right)dt.$

For this family of operators, we give here some of their properties and results.

Lemma 2

Let $n\in\mathbb{N}$ and $x\in\left[0,1\right].$ The following equalities hold.

\widetilde{K}_{n}\left(e_{0};x\right)=e^{\mu\left(x-n-1\right)/\left(n+1\right)}e^{\mu x}\left(e^{\mu/\left(n+1\right)}+1-e^{\mu x/\left(n+1\right)}\right)^{n},

(12)

\mathcal{\ }\widetilde{K}_{n}\left(\exp_{\mu};x\right)=\frac{\mu}{n+1}\frac{e^{\mu x/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1}e^{\mu x},

(13)

\mathcal{\ }\widetilde{K}_{n}\left(\exp_{\mu}^{2};x\right)=e^{2\mu x},

(14)

	$\displaystyle\mathcal{\ }\widetilde{K}_{n}\left(\exp_{\mu}^{3};x\right)$	$\displaystyle=\frac{1}{2}e^{\mu x+\mu x/\left(n+1\right)}\left(1+e^{\mu/\left(n+1\right)}\right)$		(15)
		$\displaystyle\hskip 28.45274pt\left(-e^{\mu/\left(n+1\right)}+e^{\mu x/\left(n+1\right)}+e^{\mu/\left(n+1\right)+\mu x/\left(n+1\right)}\right)^{n},$		(15)

$\displaystyle\mathcal{\ }\widetilde{K}_{n}\left(\exp_{\mu}^{4};x\right)$	$\displaystyle=\frac{1}{3}e^{\mu x+\mu x/\left(n+1\right)}\left(1+e^{\mu/\left(n+1\right)}+e^{2\mu/\left(n+1\right)}\right)$	(16)
	$\displaystyle\times\left(-e^{\mu/\left(n+1\right)}-e^{2\mu/\left(n+1\right)}+e^{\mu x/\left(n+1\right)}\right.$
	$\displaystyle\left.+e^{\mu/\left(n+1\right)+\mu x/\left(n+1\right)}+e^{2\mu/\left(n+1\right)+\mu x/\left(n+1\right)}\right)^{n}.$

Proof

It follows from (8) that if $f=e_{0},$ then

	$\displaystyle\widetilde{K}_{n}\left(e_{0};x\right)$	$\displaystyle=\frac{\mu}{n+1}\frac{e^{\mu x/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1}e^{\mu x}\left(n+1\right)\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu t}dt$
		$\displaystyle=\frac{e^{\mu x/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}}e^{\mu x}\sum\limits_{k=0}^{n}e^{-\mu k/\left(n+1\right)}p_{n,k}\left(a_{n+1}\left(x\right)\right)$
		$\displaystyle=\frac{e^{\mu x/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}}e^{\mu x}e^{-\mu n/\left(n+1\right)}\left(e^{\mu/\left(n+1\right)}+1-e^{\mu x/\left(n+1\right)}\right)^{n}$
		$\displaystyle=e^{\mu x/\left(n+1\right)}e^{\mu\left(x-1\right)}\left(e^{\mu/\left(n+1\right)}+1-e^{\mu x/\left(n+1\right)}\right)^{n}.$

If $f\left(x\right)=e^{\mu x},$ then

	$\displaystyle\widetilde{K}_{n}\left(e^{\mu t};x\right)$	$\displaystyle=a_{n+1}^{{}^{\prime}}\left(x\right)e^{\mu x}\left(n+1\right)\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}dt$
		$\displaystyle=\frac{\mu}{n+1}\frac{e^{\mu x/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1}e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)$
		$\displaystyle=\frac{\mu}{n+1}\frac{e^{\mu x/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1}e^{\mu x}.$

Finally, for $f\left(x\right)=e^{2\mu x}$ , we get

	$\displaystyle\widetilde{K}_{n}\left(e^{2\mu t};x\right)$	$\displaystyle=a_{n+1}^{{}^{\prime}}\left(x\right)e^{\mu x}\left(n+1\right)\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{\mu t}dt$
		$\displaystyle=\frac{\mu}{n+1}\frac{e^{\mu x/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1}e^{\mu x}\left(n+1\right)\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)$
		$\displaystyle\quad\cdot\left(\frac{1}{\mu}e^{\mu k/\left(n+1\right)}\left(e^{\mu/\left(n+1\right)}-1\right)\right)$
		$\displaystyle=e^{\mu x/\left(n+1\right)}e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)e^{\mu k/\left(n+1\right)}$
		$\displaystyle=e^{\mu x/\left(n+1\right)}e^{\mu x}e^{\mu xn/\left(n+1\right)}=e^{2\mu x}.$

Other results are similar.

Lemma 3

Let $\exp_{\mu,x}\left(t\right)=e^{\mu t}-e^{\mu x}.$ For $n\in\mathbb{N}$ and $x\in\left[0,1\right],$ we have

\lim_{n\rightarrow\infty}n\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right)=\mu\left(\mu x-1\right)-\mu x\left(\mu x-2\right),

(17)

\lim_{n\rightarrow\infty}n\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\right)=\frac{\mu}{2}\left(2x-1\right)e^{\mu x},

(18)

and

\lim_{n\rightarrow\infty}n^{2}\widetilde{K}_{n}\left(\exp_{\mu,x}^{4}\left(x\right);x\right)=3\mu^{2}\left(1-x\right)^{2}x^{2}e^{4\mu x}.

(19)

Using the above limits, we get

$\displaystyle\lim_{n\rightarrow\infty}n\widetilde{K}_{n}\left(\exp_{\mu,x};x\right)$	$\displaystyle=\lim_{n\rightarrow\infty}n\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\widetilde{K}_{n}\left(e_{0};x\right)\right)$
	$\displaystyle=\lim_{n\rightarrow\infty}n\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\right)$
	$\displaystyle-\lim_{n\rightarrow\infty}ne^{\mu x}\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right)$
	$\displaystyle=e^{\mu x}\left[\frac{\mu}{2}\left(2x-1\right)-\mu\left(\mu x-1\right)+\mu x\left(\mu x-2\right)\right]$	(20)

and

$\displaystyle\lim_{n\rightarrow\infty}n\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)$	$\displaystyle=-e^{\mu x}\lim_{n\rightarrow\infty}n\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\right)$
	$\displaystyle-e^{\mu x}\lim_{n\rightarrow\infty}n\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\widetilde{K}_{n}\left(e_{0};x\right)\right)$
	$\displaystyle=-2e^{\mu x}\lim_{n\rightarrow\infty}n\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\right)$
	$\displaystyle+e^{2\mu x}\lim_{n\rightarrow\infty}n\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right)$
	$\displaystyle=e^{2\mu x}\left(\mu\left(\mu x-1\right)-\mu x\left(\mu x-2\right)-\mu\left(2x-1\right)\right).$	(21)

Lemma 4

Let $\alpha_{n,\mu}:={\left\|\widetilde{K}_{n}e_{0}-e_{0}\right\|}_{\infty}$ . The following relations hold as $n\rightarrow\infty$ :

\alpha_{n,\mu}\rightarrow 0,

(22)

{\left\|\widetilde{K}_{n}(\exp_{\mu})-\exp_{\mu}\right\|}_{\infty}\rightarrow 0

(23)

and

\beta_{n,\mu}:=\sup_{x\in\left[0,1\right]}\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)\rightarrow 0.

(24)

Proof

We have

	$\displaystyle\left(\widetilde{K}_{n}e_{0}\right)^{{}^{\prime}}\left(x\right)$	$\displaystyle=\mu e^{\mu\left(x-n-1\right)/\left(n+1\right)}e^{\mu x}\left(e^{\mu/\left(n+1\right)}+1-e^{\mu x/\left(n+1\right)}\right)^{n-1}\times$
		$\displaystyle\left(\frac{\left(n+2\right)}{n+1}\left(e^{\mu/\left(n+1\right)}+1\right)-2e^{\mu x/\left(n+1\right)}\right).$

\left(\widetilde{K}_{n}e_{0}\right)^{{}^{\prime}}\left(x_{n}\right)=0,

then

x_{n}=\frac{\left(n+1\right)}{\mu}\ln\left(\frac{\left(n+2\right)}{\left(n+1\right)}\frac{\left(e^{\mu/\left(n+1\right)}+1\right)}{2}\right).

Also, we can write

	$\displaystyle\widetilde{K}_{n}e_{0}\left(0\right)-1$	$\displaystyle=e^{-\mu}e^{\mu n/\left(n+1\right)}-1$		(25)
		$\displaystyle=e^{-\mu/\left(n+1\right)}-1,$

\widetilde{K}_{n}e_{0}\left(1\right)-1=e^{\mu/\left(n+1\right)}-1

(26)

and

$\displaystyle\widetilde{K}_{n}e_{0}\left(x_{n}\right)-1$	$\displaystyle=\frac{\left(n+2\right)}{\left(n+1\right)}\frac{\left(e^{\mu/\left(n+1\right)}+1\right)}{2}\left(\frac{\left(n+2\right)}{\left(n+1\right)}\frac{\left(e^{\mu/\left(n+1\right)}+1\right)}{2}\right)^{n+1}e^{-\mu}$
	$\displaystyle\quad\times\left(e^{\mu/\left(n+1\right)}+1-\frac{\left(n+2\right)}{\left(n+1\right)}\frac{\left(e^{\mu/\left(n+1\right)}+1\right)}{2}\right)^{n}-1$
	$\displaystyle=\left(\frac{e^{\mu/\left(n+1\right)}+1}{2}\right)^{2n+2}\left(\frac{n+2}{n+1}\right)^{n+2}\left(\frac{n}{n+1}\right)^{n}e^{-\mu}-1.$	(27)

Since $\widetilde{K}_{n}e_{0}\left(x\right)-1$ is a continuous function, it attains its extreme values at either endpoints $0$ and $1$ or critical point $x_{n}$ .

Since the limits of (25), (26) and ( 27) are equal to zero, (22) is proved.

(23) and (24) follow similarly from

\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}=e^{\mu x}\left(\frac{\mu}{n+1}\frac{e^{\mu x/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1}-1\right),

and

	$\displaystyle\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)$	$\displaystyle=2e^{\mu x}\left(e^{\mu x}-\widetilde{K}_{n}\left(\exp_{\mu};x\right)\right)+e^{2\mu x}\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right)$
		$\displaystyle=2e^{2\mu x}\left(1-\frac{\mu}{n+1}\frac{e^{\mu x/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1}\right)+e^{2\mu x}\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right).$

3 Convergence in $C\left[0,1\right]$

We shall denote by $C[0,1]$ the space of all real valued continuous functions on the interval $[0,1]$ endowed with the sup-norm $\left\|\cdot\right\|_{\infty}.$

Considering the calculations in the proof of Lemma 4, we get $\left\|\widetilde{K}_{n}e_{0}\right\|_{\infty}=\alpha_{n,\mu}+1$ , where $\alpha_{n,\mu}$ is defined as in that proof. So we can say that the linear operator $\widetilde{K}_{n}$ maps $C\left[0,1\right]$ into itself and it is continuous with respect to the sup-norm. Since the sequence $\alpha_{n,\mu}$ is convergent, we get $\left\|\widetilde{K}_{n}\right\|\leq d,$ $d$ is a positive constant.

In the following theorem, we establish the convergence of the operators $\widetilde{K}_{n}$ towards the identity operator.

Theorem 3.1

If $f\in C\left[0,1\right],$ then $\widetilde{K}_{n}f$ converges to $f$ uniformly on $\left[0,1\right]$ .

Proof

Since $\left\{e_{0},\exp_{\mu},\exp_{\mu}^{2}\right\}$ is an extended complete Chebyshev system, from Korovkin’s theorem we have to prove that the thesis is fulfilled for the functions $e_{0},\exp_{\mu},\exp_{\mu}^{2}.$ But this is a consequence of Lemma 2 and Lemma 4, and so the theorem is proved.

Theorem 3.2

For every $n\in\mathbb{N}$ and $f\in C\left[0,1\right],$ we have

	$\displaystyle\left\\|\widetilde{K}_{n}f-f\right\\|_{\infty}$	$\displaystyle\leq b_{n}e^{\mu}\left\\|f\right\\|_{\infty}+2e^{\mu}\omega\left(\exp_{\mu}^{-1}f,\frac{n}{n+1}\gamma_{n}+\frac{1}{n+1}\right)+$
		$\displaystyle\quad+3e^{\mu}\omega\left(\exp_{\mu}^{-1}f,\frac{\sqrt{n}+1}{n+1}\right)$

where

b_{n}=\frac{\mu}{n+1}\frac{e^{\mu/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1}-1

and

\gamma_{n}=\frac{\left(e^{\mu/\left(n+1\right)}-1\right)\frac{\left(n+1\right)}{\mu}-1}{e^{\mu/\left(n+1\right)}-1}-\frac{\left(n+1\right)}{\mu}\ln\left[\left(e^{\mu/\left(n+1\right)}-1\right)\frac{\left(n+1\right)}{\mu}\right].

(28)

Proof

Using the representation (11) and the fact that $b_{n}=\sup_{x\in\left[0,1\right]}\!\!\left(a_{n+1}^{{}^{\prime}}(x)-1\right)=$ $=\frac{\mu}{n+1}\frac{e^{\mu/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1}-1,$ we have

	$\displaystyle\left\\|\widetilde{K}_{n}f-f\right\\|_{\infty}$	$\displaystyle\leq\sup_{x\in\left[0,1\right]}\left(a_{n+1}^{{}^{\prime}}\left(x\right)-1\right)e^{\mu}\left\\|B_{n}\left(\left(n+1\right)\delta_{n}^{\mu}\left(F\right);a_{n+1}\left(\cdot\right)\right)\right\\|_{\infty}$
		$\displaystyle\quad+\left(n+1\right)e^{\mu}\left\\|B_{n}\left(\delta_{n}^{\mu}\left(F\right);a_{n+1}\left(\cdot\right)\right)-\delta_{n}^{\mu}\left(F\right)\right\\|_{\infty}$
		$\displaystyle\quad+e^{\mu}\left\\|\left(n+1\right)\delta_{n}^{\mu}\left(F\right)-e^{-\mu\cdot}f\right\\|_{\infty}$
		$\displaystyle\leq b_{n}e^{\mu}\left\\|e^{-\mu\cdot}f\right\\|_{\infty}+\left(n+1\right)e^{\mu}\left\\|B_{n}\left(\delta_{n}^{\mu}\left(F\right);a_{n+1}\left(\cdot\right)\right)-B_{n}\left(\delta_{n}^{\mu}\left(F\right);\cdot\right)\right\\|_{\infty}$
		$\displaystyle\quad+\left(n+1\right)e^{\mu}\left\\|B_{n}\left(\delta_{n}^{\mu}\left(F\right);\cdot\right)-\delta_{n}^{\mu}\left(F\right)\right\\|_{\infty}$
		$\displaystyle\quad+e^{\mu}\left\\|\left(n+1\right)\delta_{n}^{\mu}\left(F\right)-e^{-\mu\cdot}f\right\\|_{\infty}$
		$\displaystyle:=b_{n}e^{\mu}\left\\|f\right\\|_{\infty}+I_{1}+I_{2}+I_{3}.$

To estimate $I_{1},$ we use the following inequality given by ( $6.1.96$ ) in Altomare Kitap for Bernstein operators in the case of an arbitrary continuous function on $\left[0,1\right]$ :

\omega\left(B_{n}\left(f\right),\delta\right)\leq 2\omega\left(f,\delta\right),\text{ \ \ \ }\delta>0.

Thus we have

	$\displaystyle\left\|B_{n}\left(\delta_{n}^{\mu}\left(F\right);a_{n+1}\left(x\right)\right)-B_{n}\left(\delta_{n}^{\mu}\left(F\right);x\right)\right\|$
	$\displaystyle\leq\omega\left(B_{n}\left(\delta_{n}^{\mu}\left(F\right)\right),\left\|a_{n+1}\left(x\right)-x\right\|\right)$
	$\displaystyle\leq 2\omega\left(\delta_{n}^{\mu}\left(F\right),\left\|a_{n+1}\left(x\right)-x\right\|\right)$
	$\displaystyle\leq 2\omega\left(\delta_{n}^{\mu}\left(F\right),\gamma_{n}\right),$		(29)

where

\gamma_{n}=\underset{x\in\left[0,1\right]}{\max}\left|a_{n+1}\left(x\right)-x\right|.

$a_{n+1}\left(x\right)-x$ attains its maximum value within $\left[0,1\right]$ at the point

x_{0}=\frac{n+1}{\mu}\ln\left[\left(e^{\mu/\left(n+1\right)}-1\right)\frac{\left(n+1\right)}{\mu}\right]

and thus $\gamma_{n}$ is defined as in (28). It is easily seen that

\lim_{n\rightarrow\infty}\gamma_{n}=\allowbreak 0.

Let us estimate $\omega\left(\delta_{n}^{\mu}\left(F\right),\delta\right).$ For fixed $\delta>0$ and $x,y\in\left[0,1\right],$ $\left|x-y\right|\leq\delta$ then

	$\displaystyle\left\|\delta_{n}^{\mu}\left(F\right)\left(x\right)-\delta_{n}^{\mu}\left(F\right)\left(y\right)\right\|$	$\displaystyle=\frac{1}{n+1}\left\|e^{-\mu\xi_{n,x}}f\left(\xi_{n,x}\right)-e^{-\mu\eta_{n,y}}f\left(\eta_{n,y}\right)\right\|$
		$\displaystyle\leq\frac{1}{n+1}\omega\left(\exp_{\mu}^{-1}f,\left\|\xi_{n,x}-\eta_{n,y}\right\|\right),$

where $\xi_{n,x}\in\Big[\frac{n}{n+1}x,\frac{n}{n+1}x+\frac{1}{n+1}\Big]$ and $\eta_{n,y}\in\left[\frac{n}{n+1}y,\frac{n}{n+1}y+\frac{1}{n+1}\right].$ Noting that

\left|\xi_{n,x}-\eta_{n,y}\right|\leq\frac{n}{n+1}\left|x-y\right|+\frac{1}{n+1}\leq\frac{n}{n+1}\delta+\frac{1}{n+1}

we get

\left|\delta_{n}^{\mu}\left(F\right)\left(x\right)-\delta_{n}^{\mu}\left(F\right)\left(y\right)\right|\leq\frac{1}{n+1}\omega\left(\exp_{\mu}^{-1}f,\frac{n}{n+1}\delta+\frac{1}{n+1}\right).

So it follows that

\omega\left(\delta_{n}^{\mu}\left(F\right),\delta\right)\leq\frac{1}{n+1}\omega\left(\exp_{\mu}^{-1}f,\frac{n}{n+1}\delta+\frac{1}{n+1}\right).

(30)

Using (29) and (30), we get

	$\displaystyle I_{1}$	$\displaystyle=\left(n+1\right)e^{\mu}\left\\|B_{n}\left(\delta_{n}^{\mu}\left(F\right);a_{n+1}\left(\cdot\right)\right)-B_{n}\left(\delta_{n}^{\mu}\left(F\right);\cdot\right)\right\\|_{\infty}$
		$\displaystyle\leq 2e^{\mu}\left(n+1\right)\omega\left(\delta_{n}^{\mu}\left(F\right),\gamma_{n}\right)$
		$\displaystyle\leq 2e^{\mu}\omega\left(\exp_{\mu}^{-1}f,\frac{n}{n+1}\gamma_{n}+\frac{1}{n+1}\right).$

Let us estimate $I_{2}$ . Using (30), we have

	$\displaystyle I_{2}$	$\displaystyle=e^{\mu}\left(n+1\right)\left\\|B_{n}\left(\delta_{n}^{\mu}\left(F\right);\cdot\right)-\delta_{n}^{\mu}\left(F\right)\right\\|_{\infty}$
		$\displaystyle\leq 2e^{\mu}\left(n+1\right)\omega\left(\delta_{n}^{\mu}\left(F\right),\frac{1}{\sqrt{n}}\right)$
		$\displaystyle\leq 2e^{\mu}\omega\left(\exp_{\mu}^{-1}f,\frac{\sqrt{n}+1}{n+1}\right).$

Now we proceed to estimate $I_{3}.$ For $x\in\left[0,1\right],$ by using Lagrange’s theorem we have

	$\displaystyle\left(n+1\right)\delta_{n}^{\mu}\left(F\right)\left(x\right)$	$\displaystyle=\left(n+1\right)\left[F_{\mu}\left(\frac{n}{n+1}x+\frac{1}{n+1}\right)-F_{\mu}\left(\frac{n}{n+1}x\right)\right]$
		$\displaystyle=F_{\mu}^{{}^{\prime}}\left(\xi_{n,x}\right)=e^{-\mu\xi_{n,x}}f\left(\xi_{n,x}\right),$		(31)

where $\xi_{n,x}\in\left[\frac{n}{n+1}x,\frac{n}{n+1}x+\frac{1}{n+1}\right].$ Since $\left|\xi_{n,x}-x\right|<\frac{1}{n+1}$ we get

	$\displaystyle\left\|\left(n+1\right)\delta_{n}^{\mu}\left(F\right)\left(x\right)-e^{-\mu x}f\left(x\right)\right\|$	$\displaystyle\leq\left\|e^{-\mu\xi_{n,x}}f\left(\xi_{n,x}\right)-e^{-\mu x}f\left(x\right)\right\|$
		$\displaystyle\leq\omega\left(\exp_{\mu}^{-1}f,\frac{1}{n+1}\right)$

and so

I_{3}=e^{\mu}\left\|\left(n+1\right)\delta_{n}^{\mu}\left(F\right)-e^{-\mu\cdot}f\right\|_{\infty}\leq e^{\mu}\omega\left(\exp_{\mu}^{-1}f,\frac{1}{n+1}\right).

Collecting all the estimates of $I_{1},$ $I_{2\text{ }}$ and $I_{3},$ we have the desired result.

Remark 1

It is easy to verify that the sequences $b_{n}$ and $\gamma_{n}$ in Theorem 3.2 satisfy $b_{n}={\cal O}(\tfrac{1}{n})$ and $\gamma_{n}={\cal O}(\tfrac{1}{n})$ as $n\rightarrow\infty$ .

4 Quantitative Voronovskaya theorem

To describe the rate of pointwise convergence of the operators, we give a quantitative version of Voronovskaya theorem. Note that for a general linear positive operators similar results were given in Go-Pi-Ra .

For $f\in C^{2}\left[0,1\right]$ and $x_{0},$ $x\in\left[0,1\right],$ we use the following version of the Taylor formula:

	$\displaystyle f\left(x\right)$	$\displaystyle=\left(f\circ\log_{\mu}\right)\left(e^{\mu x_{0}}\right)+\left(f\circ\log_{\mu}\right)^{{}^{\prime}}\left(e^{\mu x_{0}}\right)\exp_{\mu,x_{0}}\left(x\right)+$
		$\displaystyle\quad+\frac{1}{2}\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\left(e^{\mu x_{0}}\right)\exp_{\mu,x_{0}}^{2}\left(x\right)+R\left(f;x_{0};x\right),$

where $\log_{\mu}$ is the inverse function of $e^{\mu}$ and the remainder $R\left(f;x_{0};x\right)$ is

R\left(f;x_{0};x\right)=\frac{1}{2}\exp_{\mu,x_{0}}^{2}\left(x\right)\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\left(e^{\mu\xi_{x}}\right)-\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\left(e^{\mu x_{0}}\right)\right),

with $\xi_{x}$ between $x$ and $x_{0}.$ Thus, we can write

$\displaystyle\left\|R\left(f,x_{0};x\right)\right\|$	$\displaystyle=\frac{1}{2}\exp_{\mu,x_{0}}^{2}\left(x\right)\left\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\left(e^{\mu\xi_{x}}\right)-\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\left(e^{\mu x}\right)\right\|$
	$\displaystyle\leq\frac{1}{2}\exp_{\mu,x_{0}}^{2}\left(x\right)\omega\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\left\|\exp_{\mu,x}\left(\xi_{x}\right)\right\|\right)$
	$\displaystyle\leq\frac{1}{2}\exp_{\mu,x_{0}}^{2}\left(x\right)\omega\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\left\|\exp_{\mu,x}\left(x_{0}\right)\right\|\right),$	(32)

where $\omega\left(f;\cdot\right)$ is the classical modulus of continuity.

We will need the following $K$ -functional introduced by J. Peetre petre and defined by

K\left(f;\varepsilon,C,C^{1}\right):=\inf\left\{\left\|f-g\right\|_{\infty}+\varepsilon\left\|g^{{}^{\prime}}\right\|_{\infty}:g\in C^{1}\right\}

(33)

for $f\in C\left[0,1\right]$ and $\varepsilon\geq 0.$ The $K$ -functional and the modulus of continuity are related by the relation

K\left(f;\varepsilon/2,C,C^{1}\right)=\frac{1}{2}\widetilde{\omega}\left(f;\varepsilon\right).

(34)

Here $\widetilde{\omega}\left(f;\cdot\right)$ denotes the least concave majorant of $\omega\left(f;\cdot\right),$ see Lorentz .

The remainder $R\left(f,x_{0};x\right)$ can be estimated in terms of $\widetilde{\omega}.$

Lemma 5

Let $f\in C^{2}\left[0,1\right]$ and $x_{0},$ $x\in\left[0,1\right].$ Then we have

\left|R\left(f,x_{0};x\right)\right|\leq\frac{1}{2}\exp_{\mu,x_{0}}^{2}\left(x\right)\widetilde{\omega}\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\frac{1}{3}\left|\exp_{\mu,x_{0}}\left(x\right)\right|\right).

Proof

From (32),we have

\left|R\left(f,x_{0};x\right)\right|\leq\exp_{\mu,x_{0}}^{2}\left(x\right)\left\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\right\|_{\infty}.

For $g\in C^{3}\left[0,1\right],$ using the Lagrange form of remainder we have

	$\displaystyle\left\|R\left(g,x_{0};x\right)\right\|$	$\displaystyle=\frac{1}{6}\left\|\exp_{\mu,x_{0}}^{3}\left(x\right)\right\|\left\|\left(g\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime\prime}}}\left(\theta_{x}\right)\right\|$
		$\displaystyle\leq\frac{1}{6}\left\|\exp_{\mu,x_{0}}^{3}\left(x\right)\right\|\left\\|\left(g\circ\log_{\mu}\right)^{{}^{{}^{{}^{\prime\prime\prime}}}}\right\\|_{\infty},$

where $\theta_{x}$ is between $x$ and $x_{0}.$ Keeping $f$ fixed and letting $g$ arbitrary in $C^{3}\left[0,1\right],$ we have

	$\displaystyle\left\|R\left(f,x_{0};x\right)\right\|$	$\displaystyle\leq\left\|R\left(f-g,x_{0};x\right)\right\|+\left\|R\left(g,x_{0};x\right)\right\|$
		$\displaystyle\leq\exp_{\mu,x_{0}}^{2}\left(x\right)\left\\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}-\left(g\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\right\\|_{\infty}$
		$\displaystyle\quad+\frac{1}{6}\left\|\exp_{\mu,x_{0}}^{3}\left(x\right)\right\|\left\\|\left(g\circ\log_{\mu}\right)^{{}^{{}^{{}^{\prime\prime\prime}}}}\right\\|_{\infty}.$

Considering (33), (34) and passing to infimum over $g\in C^{3}\left[0,1\right]$ yields

	$\displaystyle\left\|R\left(f,x_{0};x\right)\right\|$	$\displaystyle\leq\exp_{\mu,x_{0}}^{2}\left(x\right)K\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\frac{1}{6}\left\|\exp_{\mu,x_{0}}(x)\right\|,C,C^{1}\right)$
		$\displaystyle=\frac{1}{2}\exp_{\mu,x_{0}}^{2}\left(x\right)\widetilde{\omega}\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\frac{1}{3}\left\|\exp_{\mu,x_{0}}\left(x\right)\right\|\right).$

Theorem 4.1

If $f\in C^{2}\left[0,1\right]$ and $x\in\left[0,1\right],$ then

	$\displaystyle\left\|\widetilde{K}_{n}\left(f;x\right)-f\left(x\right)-\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right)\frac{1}{2\mu^{2}}\left[f^{{}^{\prime\prime}}\left(x\right)-3\mu f^{{}^{\prime}}\left(x\right)+2\mu^{2}f\left(x\right)\right]\right.$
	$\displaystyle\left.-e^{-\mu x}\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\right)\frac{1}{\mu^{2}}\left[2\mu f^{{}^{\prime}}\left(x\right)-f^{{}^{\prime\prime}}\left(x\right)\right]\right\|$
	$\displaystyle\leq\frac{1}{2}\sqrt{\widetilde{K}_{n}\left(\exp_{\mu,x}^{4};x\right)}\sqrt{\widetilde{K}_{n}\left(e_{0};x\right)}\widetilde{\omega}\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\frac{1}{3}\frac{\sqrt{\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)}}{\sqrt{\widetilde{K}_{n}\left(e_{0};x\right)}}\right).$

Proof

For $f\in C^{2}\left[0,1\right],$ we can write

R\left(f,x;t\right)=f\left(t\right)-\left(f\circ\log_{\mu}\right)\left(e^{\mu x}\right)-\left(f\circ\log_{\mu}\right)^{{}^{\prime}}\left(e^{\mu x}\right)\exp_{\mu,x}\left(t\right)-\frac{1}{2}\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\left(e^{\mu x}\right)\exp_{\mu,x}^{2}\left(t\right).

This equality leads to

	$\displaystyle\widetilde{K}_{n}\left(R\left(f,x;t\right);x\right)$	$\displaystyle=\widetilde{K}_{n}\left(f;x\right)-f\left(x\right)\widetilde{K}_{n}\left(e_{0};x\right)-\left(f\circ\log_{\mu}\right)^{{}^{\prime}}\left(e^{\mu x}\right)\widetilde{K}_{n}\left(\exp_{\mu,x};x\right)$
		$\displaystyle\text{\ }-\frac{1}{2}\left(f\circ\log_{\mu}\right)^{{}^{\prime\prime}}\left(e^{\mu x}\right)\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)\text{.}$

Since

\left(f\circ\log_{\mu}\right)^{{}^{\prime}}\left(e^{\mu x}\right)=\frac{e^{-\mu x}}{\mu}f^{{}^{\prime}}\left(x\right)

and

\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\left(e^{\mu x}\right)=e^{-2\mu x}\left(\frac{1}{\mu^{2}}f^{{}^{\prime\prime}}\left(x\right)-\frac{1}{\mu}f^{{}^{\prime}}\left(x\right)\right),

we can write

	$\displaystyle\widetilde{K}_{n}\left(R\left(f,x;t\right);x\right)$	$\displaystyle=\widetilde{K}_{n}\left(f;x\right)-f\left(x\right)\widetilde{K}_{n}\left(e_{0};x\right)-\frac{e^{-\mu x}}{\mu}f^{{}^{\prime}}\left(x\right)\widetilde{K}_{n}\left(\exp_{\mu,x};x\right)$
		$\displaystyle-\frac{e^{-2\mu x}}{2}\left(\frac{1}{\mu^{2}}f^{{}^{\prime\prime}}\left(x\right)-\frac{1}{\mu}f^{{}^{\prime}}\left(x\right)\right)\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right).$

It can be rearranged as

	$\displaystyle\widetilde{K}_{n}\left(R\left(f,x;t\right);x\right)=$
	$\displaystyle=\widetilde{K}_{n}\left(f;x\right)-f\left(x\right)-f\left(x\right)\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right)$
	$\displaystyle\quad-\frac{1}{\mu}f^{{}^{\prime}}\left(x\right)\left[e^{-\mu x}\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\right)-\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right)\right]$
	$\displaystyle\quad-\frac{1}{2}\left(\frac{1}{\mu^{2}}f^{{}^{\prime\prime}}\left(x\right)-\frac{1}{\mu}f^{{}^{\prime}}\left(x\right)\right)\left[\left(\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right)-2e^{-\mu x}\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\right)\right)\right]$
	$\displaystyle=\widetilde{K}_{n}\left(f;x\right)-f\left(x\right)-\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right)\left[f\left(x\right)+\frac{1}{2}\left(\frac{1}{\mu^{2}}f^{{}^{\prime\prime}}\left(x\right)-\frac{1}{\mu}f^{{}^{\prime}}\left(x\right)\right)-\frac{1}{\mu}f^{{}^{\prime}}\left(x\right)\right]$
	$\displaystyle\quad-e^{-\mu x}\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\right)\left[\frac{1}{\mu}f^{{}^{\prime}}\left(x\right)-\left(\frac{1}{\mu^{2}}f^{{}^{\prime\prime}}\left(x\right)-\frac{1}{\mu}f^{{}^{\prime}}\left(x\right)\right)\right]$
	$\displaystyle=\widetilde{K}_{n}\left(f;x\right)-f\left(x\right)-\left(\widetilde{K}_{n}\left(e_{0};x\right)-1\right)\frac{1}{2\mu^{2}}\left[f^{{}^{\prime\prime}}\left(x\right)-3\mu f^{{}^{\prime}}\left(x\right)+2\mu^{2}f\left(x\right)\right]$
	$\displaystyle\quad-e^{-\mu x}\left(\widetilde{K}_{n}\left(\exp_{\mu};x\right)-e^{\mu x}\right)\frac{1}{\mu^{2}}\left[2\mu f^{{}^{\prime}}\left(x\right)-f^{{}^{\prime\prime}}\left(x\right)\right].$

On the other hand, from Lemma 5 we have

\left|\widetilde{K}_{n}\left(R\left(f,x;t\right);x\right)\right|\leq\widetilde{K}_{n}\left(\frac{1}{2}\exp_{\mu,x}^{2}\left(t\right)\widetilde{\omega}\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\frac{1}{3}\left|\exp_{\mu,x}\left(t\right)\right|\right);x\right).

For arbitrary $g\in C^{3}\left[0,1\right],$ we get

	$\displaystyle\left\|\widetilde{K}_{n}\left(R\left(f,x;t\right);x\right)\right\|=$
	$\displaystyle=\widetilde{K}_{n}\left(\exp_{\mu,x}^{2}\left(t\right)K\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\frac{1}{6}\left\|\exp_{\mu,x}\left(t\right)\right\|,C,C^{1}\right);x\right)$
	$\displaystyle\leq\widetilde{K}_{n}\left(\exp_{\mu,x}^{2}\left(t\right)\left\{\left\\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}-\left(g\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\right\\|_{\infty}+\frac{\left\|\exp_{\mu,x}\left(t\right)\right\|}{6}\left\\|\left(g\circ\log_{\mu}\right)^{{}^{{}^{{}^{\prime\prime\prime}}}}\right\\|_{\infty}\right\};x\right)$
	$\displaystyle=\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)\left\\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}-\left(g\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\right\\|_{\infty}+\frac{1}{6}\widetilde{K}_{n}\left(\left\|\exp_{\mu,x}^{3}\right\|;x\right)\left\\|\left(g\circ\log_{\mu}\right)^{{}^{{}^{{}^{\prime\prime\prime}}}}\right\\|_{\infty}$
	$\displaystyle\leq\sqrt{\widetilde{K}_{n}\left(\exp_{\mu,x}^{4};x\right)}\sqrt{\widetilde{K}_{n}\left(e_{0};x\right)}\left\{\left\\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}-\left(g\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\right\\|_{\infty}\right.$
	$\displaystyle\quad+\frac{1}{6}\frac{\sqrt{\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)}}{\sqrt{\widetilde{K}_{n}\left(e_{0};x\right)}}\left\\|\left(g\circ\log_{\mu}\right)^{{}^{{}^{{}^{\prime\prime\prime}}}}\right\\|_{\infty}\Bigg\}.$

Passing to the infimum over $g\in C^{3}\left[0,1\right]$ again, we get

	$\displaystyle\left\|\widetilde{K}_{n}\left(R\left(f,x;t\right);x\right)\right\|\leq$
	$\displaystyle\leq\sqrt{\widetilde{K}_{n}\left(\exp_{\mu,x}^{4};x\right)}\sqrt{\widetilde{K}_{n}\left(e_{0};x\right)}K\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\frac{1}{6}\frac{\sqrt{\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)}}{\sqrt{\widetilde{K}_{n}\left(e_{0};x\right)}},C,C^{1}\right)$
	$\displaystyle=\frac{1}{2}\sqrt{\widetilde{K}_{n}\left(\exp_{\mu,x}^{4};x\right)}\sqrt{\widetilde{K}_{n}\left(e_{0};x\right)}\widetilde{\omega}\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\frac{1}{3}\frac{\sqrt{\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)}}{\sqrt{\widetilde{K}_{n}\left(e_{0};x\right)}}\right).$

Using (17), (18), (21) and (19) in Theorem 4.1, respectively, and considering

\lim_{n\rightarrow\infty}\frac{\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)}{\widetilde{K}_{n}\left(e_{0};x\right)}=0

we have:

Corollary 1

If $f\in C^{2}\left[0,1\right]$ and $x\in\left[0,1\right],$ then

	$\displaystyle\lim_{n\rightarrow\infty}2n\mu^{2}\left(\widetilde{K}_{n}\left(f;x\right)-f\left(x\right)\right)=$
	$\displaystyle=\left(\mu\left(\mu x-1\right)-\mu x\left(\mu x-2\right)\right)\left[f^{{}^{\prime\prime}}\left(x\right)-3\mu f^{{}^{\prime}}\left(x\right)+2\mu^{2}f\left(x\right)\right]$
	$\displaystyle\quad+\mu\left(2x-1\right)\left[2\mu f^{{}^{\prime}}\left(x\right)-f^{{}^{\prime\prime}}\left(x\right)\right].$

5 Convergence in $L_{p,\mu}\left[0,1\right]$ and $L_{p}\left[0,1\right]$

Let $1\leq p<\infty$ be fixed and $L_{p,\mu}\left[0,1\right]$ be the space of all functions for which $\exp_{\mu}f$ is Lebesgue integrable with the $p$ -power over $\left[0,1\right].$ The norm in $L_{p,\mu}\left[0,1\right]$ is defined as

\left\|f\right\|_{p,\mu}:=\left(\int_{0}^{1}\left|e^{-\mu x}f\left(x\right)\right|^{p}dx\right)^{1/p}.

The norm of a linear operator $L_{n}$ acting from the space $L_{p,\mu}$ to $L_{p,\mu}$ given by

\left\|L_{n}\right\|_{L_{p,\mu},L_{p,\mu}}:=\underset{\left\|f\right\|_{p,\mu}\neq 0}{\sup}\frac{\left\|L_{n}f\right\|_{p,\mu}}{\left\|f\right\|_{p,\mu}}.

Also, $\widetilde{K}_{n}$ maps the space $L_{1,\mu}\left[0,1\right]$ into the space $C\left[0,1\right].$

Lemma 6

For $f\in L_{p,\mu}\left[0,1\right],$ we have $\mathcal{\ }\widetilde{K}_{n}\left(f\right)\in L_{p,\mu}\left[0,1\right]$ and for all $n\in\mathbb{N},$ $\left\|\widetilde{K}_{n}\right\|_{L_{p,\mu},L_{p,\mu}}\leq e^{\mu\frac{\left(p-1\right)}{p}}.$

Proof

Applying the Jensen’s inequality to the measure $(n+1)dt$ , we get

$\displaystyle\left\|\widetilde{K}_{n}\left(f;x\right)\right\|^{p}$	$\displaystyle\leq\left(\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)e^{\mu x}\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu t}\left\|f\left(t\right)\right\|dt\right)^{p}$
	$\displaystyle\leq\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\left(a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)e^{\mu x}\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu t}\left\|f\left(t\right)\right\|dt\right)^{p}$
	$\displaystyle\leq\max_{x\in\left[0,1\right]}\left(a_{n+1}^{{}^{\prime}}\left(x\right)\right)^{p-1}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)a_{n+1}^{{}^{\prime}}\left(x\right)e^{\mu px}\left(n+1\right)$
	$\displaystyle\quad\cdot\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu pt}\left\|f\left(t\right)\right\|^{p}dt.$	(35)

If we take the integral on the interval $\left[0,1\right]$ , we obtain

	$\displaystyle\int_{0}^{1}\left\|e^{-\mu x}\widetilde{K}_{n}\left(f;x\right)\right\|^{p}dx$
	$\displaystyle\leq\max_{x\in\left[0,1\right]}\left(a_{n+1}^{{}^{\prime}}\left(x\right)\right)^{p-1}\sum\limits_{k=0}^{n}\int_{0}^{1}p_{n,k}\left(a_{n+1}\left(x\right)\right)a_{n+1}^{{}^{\prime}}\left(x\right)dx$
	$\displaystyle\quad\cdot\left[\left(n+1\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu pt}\left\|f\left(t\right)\right\|^{p}dt\right]$
	$\displaystyle=\max_{x\in\left[0,1\right]}\left(a_{n+1}^{{}^{\prime}}\left(x\right)\right)^{p-1}\sum\limits_{k=0}^{n}\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu pt}\left\|f\left(t\right)\right\|^{p}dt$
	$\displaystyle=\max_{x\in\left[0,1\right]}\left(a_{n+1}^{{}^{\prime}}\left(x\right)\right)^{p-1}\int_{0}^{1}e^{-\mu pt}\left\|f\left(t\right)\right\|^{p}dt.$

Since

\max_{x\in\left[0,1\right]}\left(a_{n+1}^{{}^{\prime}}\left(x\right)\right)=\frac{\mu}{n+1}\frac{e^{\mu/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1},

using the inequality $u\leq e^{u}-1,$ $u>0,$ we have

	$\displaystyle\max_{x\in\left[0,1\right]}\left(a_{n+1}^{{}^{\prime}}\left(x\right)\right)$	$\displaystyle=\frac{\mu}{n+1}\frac{e^{\mu/\left(n+1\right)}}{e^{\mu/\left(n+1\right)}-1}\leq\frac{\mu}{n+1}\left(\frac{e^{\mu/\left(n+1\right)}}{\mu/\left(n+1\right)}\right)$
		$\displaystyle\leq e^{\mu/\left(n+1\right)}.$		(36)

Consequently we get

\left\|\widetilde{K}_{n}\left(f\right)\right\|_{p,\mu}\leq e^{\frac{\mu}{n+1}\frac{\left(p-1\right)}{p}}\left\|f\right\|_{p,\mu}.

In view of this inequality, for all $n\in\mathbb{N}$ we can write

\left\|\widetilde{K}_{n}\right\|_{L_{p,\mu},L_{p,\mu}}\leq e^{\mu\frac{\left(p-1\right)}{p}}.

Theorem 5.1

Let $f\in L_{p,\mu}\left[0,1\right]$ for $1\leq p<\infty.$ Then we have

\lim_{n\rightarrow\infty}\left\|\widetilde{K}_{n}\left(f\right)-f\right\|_{p,\mu}=0.

Proof

From the Luzin theorem for a given $\varepsilon>0$ , there exists $g\in C[0,1]$ such that

\left\|f-g\right\|_{p,\mu}<\varepsilon.

On the other hand, since $\widetilde{K}_{n}g$ converges to $g$ uniformly on $\left[0,1\right],$ there exists an $n_{0}\in N$ such that, for $n\geq n_{0}$

\left\|\widetilde{K}_{n}\left(g\right)-g\right\|_{\infty}<\varepsilon.

In view of the above inequalities

	$\displaystyle\left\\|\widetilde{K}_{n}\left(f\right)-f\right\\|_{p,\mu}$	$\displaystyle\leq\left\\|\widetilde{K}_{n}\left(f\right)-\widetilde{K}_{n}\left(g\right)\right\\|_{p,\mu}+\left\\|\widetilde{K}_{n}\left(g\right)-g\right\\|_{\infty}+\left\\|f-g\right\\|_{p,\mu}$
		$\displaystyle\leq\left(\left\\|\widetilde{K}_{n}\right\\|_{L_{p,\mu},L_{p,\mu}}+1\right)\left\\|f-g\right\\|_{p,\mu}+\left\\|\widetilde{K}_{n}\left(g\right)-g\right\\|_{\infty}$
		$\displaystyle\leq\left(\left\\|\widetilde{K}_{n}\right\\|_{L_{p,\mu},L_{p,\mu}}+2\right)\varepsilon.$

for $n\geq n_{0}.$ Thus, we have the desired result.

Now we want to give a quantitative approximation theorem for (8) in the space $L_{p}\left[0,1\right].$ To describe our results we will use the following integral modulus of continuity defined by

\omega\left(f;t\right)_{L_{p}}=\sup_{0\leq h<t}\left\|f\left(\cdot+h\right)-f\left(\cdot\right)\right\|_{L_{p}\left[0,1\right].}

$\left\|\cdot\right\|_{L_{p}}$ denotes the usual $L_{p}$ norm and the corresponding $K$ -functionals is defined by

K_{p}\left(f;t\right):=\inf_{g\in AC\left[0,1\right],g^{{}^{\prime}}\in L_{p}}\left\{\left\|f-g\right\|_{L_{p}}+t\left\|g^{{}^{\prime}}\right\|_{L_{p}}\right\},

where $AC\left[0,1\right]$ indicates the set of all absolute continuous functions on the interval $\left[0,1\right]$ . We know that $K_{p}\left(f;t\right)$ and $\omega\left(f;t\right)_{L_{p}}$ are equivalent (see (Devore, , Theorem 2.1)), i.e., there is a constant $C$ such that

C^{-1}\omega\left(f;t\right)_{L_{p}}\leq K_{p}\left(f;t\right)\leq C\omega\left(f;t\right)_{L_{p}}.

(37)

For given $f\in L_{p}\left[0,1\right]$ , the Hardy-Littlewood maximum function is defined by

M\left(f;x\right)=\underset{t\neq x}{\sup_{0\leq t\leq 1}}\frac{1}{t-x}\int_{x}^{t}\left|f\left(u\right)\right|du.

(38)

It is well known that (see Stein )

\left\|M\left(f\right)\right\|_{L_{p}}\leq C_{p}\left\|f\right\|_{L_{p}}.

(39)

Also, as in (35), considering (36) we can write

	$\displaystyle\left\|\widetilde{K}_{n}\left(f;x\right)\right\|^{p}$	$\displaystyle\leq\max_{x\in\left[0,1\right]}\left(a_{n+1}^{{}^{\prime}}\left(x\right)\right)^{p-1}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)a_{n+1}^{{}^{\prime}}\left(x\right)e^{\mu px}\left(n+1\right)$
		$\displaystyle\quad\cdot\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu pt}\left\|f\left(t\right)\right\|^{p}dt$
		$\displaystyle\leq e^{\mu p}e^{\frac{\mu}{n+1}\left(p-1\right)}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}\left\|f\left(t\right)\right\|^{p}dt.$

Integrating, we have

	$\displaystyle\int_{0}^{1}\left\|\widetilde{K}_{n}\left(f;x\right)\right\|^{p}dx$	$\displaystyle\leq e^{\mu p}e^{\frac{\mu}{n+1}\left(p-1\right)}\sum\limits_{k=0}^{n}\left(\left(n+1\right)\int_{0}^{1}p_{n,k}\left(a_{n+1}\left(x\right)\right)a_{n+1}^{{}^{\prime}}\left(x\right)dx\right)$
		$\displaystyle\quad\cdot\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}\left\|f\left(t\right)\right\|^{p}dt$
		$\displaystyle\leq e^{\mu p}e^{\frac{\mu}{n+1}\left(p-1\right)}\sum\limits_{k=0}^{n}\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}\left\|f\left(t\right)\right\|^{p}dt$

and

\left\|\widetilde{K}_{n}\left(f\right)\right\|_{p}\leq e^{\mu}e^{\frac{\mu}{n+1}\frac{\left(p-1\right)}{p}}\left\|f\right\|_{p}.

Consequently, for all $n\in\mathbb{N}$ we can write

\left\|\widetilde{K}_{n}\right\|_{L_{p},L_{p}}\leq e^{\mu}e^{\mu\frac{\left(p-1\right)}{p}}.

(40)

Lemma 7

Let $g\in AC[0,1]$ and $g^{{}^{\prime}}\in L_{p}[0,1],$ $p>1.$ Then we have

\left\|\widetilde{K}_{n}\left(g\right)-g\right\|_{p}\leq\alpha_{n,\mu}\left\|g\right\|_{p}+\mu^{-1}\left(\beta_{n,\mu}\right)^{1/2}e^{\frac{\mu}{2}\left(2+\frac{1}{n+1}\right)}C_{p}\left\|g^{{}^{\prime}}\right\|_{p}.

where $\alpha_{n,\mu}$ , $\beta_{n,\mu}$ are defined as in Lemma 4 and $C_{p}$ is defined in (39).

Proof

Using the equality $g\left(t\right)=g\left(x\right)+\int_{x}^{t}g^{{}^{\prime}}\left(u\right)du,$ we can write

	$\displaystyle\widetilde{K}_{n}\left(g;x\right)-g\left(x\right)+g\left(x\right)-g\left(x\right)\widetilde{K}_{n}\left(e_{0};x\right)=$
	$\displaystyle=a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu t}\left(g\left(t\right)-g\left(x\right)\right)dt$
	$\displaystyle=a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}e^{-\mu t}\int_{x}^{t}g^{{}^{\prime}}\left(u\right)dudt.$

Hence

	$\displaystyle\left\|\widetilde{K}_{n}\left(g;x\right)-g\left(x\right)\right\|\leq$
	$\displaystyle\leq\max_{x\in\left[0,1\right]}\left\|\widetilde{K}_{n}\left(e_{0};x\right)-1\right\|\left\|g\left(x\right)\right\|$
	$\displaystyle\quad+M\left(g^{{}^{\prime}};x\right)a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}\left\|t-x\right\|dt,$

where

M\left(g^{{}^{\prime}};x\right)=\underset{x\neq t}{\sup_{0\leq t\leq 1}}\frac{1}{t-x}\int_{x}^{t}\left|g^{{}^{\prime}}\left(u\right)\right|du

is defined as in (38). By Lagrange’s theorem for $x\in\left(0,1\right)$ and $t\in\left[0,1\right],$ we get

\mu\left|t-x\right|\leq\left|\exp_{\mu,x}\left(t\right)\right|.

Considering (22) and using the above inequality with Cauchy-Schwarz inequality, we get

	$\displaystyle\left\|\widetilde{K}_{n}\left(g;x\right)-g\left(x\right)\right\|\leq$
	$\displaystyle\leq\alpha_{n,\mu}\left\|g\left(x\right)\right\|+\mu^{-1}M\left(g^{{}^{\prime}};x\right)\left(a_{n+1}^{{}^{\prime}}\left(x\right)\left(n+1\right)e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}dt\right)^{1/2}$
	$\displaystyle\quad\times\left(a_{n+1}^{{}^{\prime}}\left(x\right)e^{\mu x}\left(n+1\right)\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}\exp_{\mu,x}^{2}\left(t\right)dt\right)^{1/2}$
	$\displaystyle\leq\alpha_{n,\mu}\left\|g\left(x\right)\right\|$
	$\displaystyle\quad+\mu^{-1}M\left(g^{{}^{\prime}};x\right)\left(a_{n+1}^{{}^{\prime}}\left(x\right)e^{\mu x}\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\right)^{1/2}$
	$\displaystyle\times\left(a_{n+1}^{{}^{\prime}}\left(x\right)e^{\mu x}\left(n+1\right)\sum\limits_{k=0}^{n}p_{n,k}\left(a_{n+1}\left(x\right)\right)\int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}}\exp_{\mu,x}^{2}\left(t\right)dt\right)^{1/2}.$

Using (36) we have

\left|\widetilde{K}_{n}\left(g;x\right)-g\left(x\right)\right|\leq\alpha_{n,\mu}\left|g\left(x\right)\right|+\mu^{-1}M\left(g^{{}^{\prime}};x\right)e^{\frac{\mu}{2}\left(2+\frac{1}{n+1}\right)}\left(\widetilde{K}_{n}\left(\exp_{\mu,x}^{2}\left(t\right);x\right)\right)^{1/2}.

From (24) we can write

\left|\widetilde{K}_{n}\left(g;x\right)-g\left(x\right)\right|\leq\alpha_{n,\mu}\left|g\left(x\right)\right|+\mu^{-1}M\left(g^{{}^{\prime}};x\right)e^{\frac{\mu}{2}\left(2+\frac{1}{n+1}\right)}\left(\beta_{n,\mu}\right)^{1/2}.

Considering (39), we get

\left\|\widetilde{K}_{n}\left(g\right)-g\right\|_{p}\leq\alpha_{n,\mu}\left\|g\right\|_{p}+\mu^{-1}\left(\beta_{n,\mu}\right)^{1/2}e^{\frac{\mu}{2}\left(2+\frac{1}{n+1}\right)}C_{p}\left\|g^{{}^{\prime}}\right\|_{p}.

Theorem 5.2

Let $f\in L_{p}\left[0,1\right],$ $p>1.$ Then we have

\left\|\widetilde{K}_{n}\left(f\right)-f\right\|_{p}\leq\alpha_{n,\mu}\left\|f\right\|_{p}+CK\omega\left(f;\left(\beta_{n,\mu}\right)^{1/2}\right)_{L_{p}},

where $K:=\max\left\{e^{\mu}e^{\mu\frac{\left(p-1\right)}{p}}+\alpha_{n,\mu}+1,\mu^{-1}e^{\frac{\mu}{2}\left(2+\frac{1}{n+1}\right)}C_{p}\right\}.$

Proof

For $g\in AC\left[0,1\right],\ g^{\prime}\in L_{p}$ , we can write

	$\displaystyle\left\\|\widetilde{K}_{n}\left(f\right)-f\right\\|_{p}$	$\displaystyle=\left\\|\widetilde{K}_{n}\left(f-g+g\right)-f\right\\|_{p}$
		$\displaystyle\leq\left\\|\widetilde{K}_{n}\left(f-g\right)-\left(f-g\right)+\left(\widetilde{K}_{n}\left(g\right)-g\right)\right\\|_{p}$
		$\displaystyle\leq\left\\|\widetilde{K}_{n}\left(f-g\right)\right\\|_{p}+\left\\|f-g\right\\|_{p}+\left\\|\widetilde{K}_{n}\left(g\right)-g\right\\|_{p}$
		$\displaystyle\leq\left(\left\\|\widetilde{K}_{n}\right\\|_{L_{p},L_{p}}+1\right)\left\\|f-g\right\\|_{p}+\left\\|\widetilde{K}_{n}\left(g\right)-g\right\\|_{p}$

Using the fact that $\left\|\widetilde{K}_{n}\right\|_{L_{p},L_{p}}\leq e^{\mu}e^{\mu\frac{\left(p-1\right)}{p}}$ (see (40)) and Lemma 7, we get

	$\displaystyle\left\\|\widetilde{K}_{n}\left(f\right)-f\right\\|_{p}\leq$
	$\displaystyle\leq\left(\left\\|\widetilde{K}_{n}\right\\|_{L_{p},L_{p}}+1\right)\left\\|f-g\right\\|_{p}+\alpha_{n,\mu}\left\\|g\right\\|_{p}+\mu^{-1}\left(\beta_{n,\mu}\right)^{1/2}e^{\frac{\mu}{2}\left(2+\frac{1}{n+1}\right)}C_{p}\left\\|g^{{}^{\prime}}\right\\|_{p}$
	$\displaystyle\leq\left(e^{\mu}e^{\mu\frac{\left(p-1\right)}{p}}+1\right)\left\\|f-g\right\\|_{p}+\alpha_{n,\mu}\left\\|f-g\right\\|_{p}+\alpha_{n,\mu}\left\\|f\right\\|_{p}+$
	$\displaystyle\quad+\mu^{-1}\left(\beta_{n,\mu}\right)^{1/2}e^{\frac{\mu}{2}\left(2+\frac{1}{n+1}\right)}C_{p}\left\\|g^{{}^{\prime}}\right\\|_{p}.$

Define $\max\left\{e^{\mu}e^{\mu\frac{\left(p-1\right)}{p}}+\alpha_{n,\mu}+1,\mu^{-1}e^{\frac{\mu}{2}\left(2+\frac{1}{n+1}\right)}C_{p}\right\}:=K$ , we have (see (5.3))

	$\displaystyle\left\\|\widetilde{K}_{n}\left(f\right)-f\right\\|_{p}$	$\displaystyle\leq\alpha_{n,\mu}\left\\|f\right\\|_{p}+K\left(\left\\|f-g\right\\|_{p}+\left(\beta_{n,\mu}\right)^{1/2}\left\\|g^{{}^{\prime}}\right\\|_{p}\right)$
		$\displaystyle\leq\alpha_{n,\mu}\left\\|f\right\\|_{p}+CK\omega\left(f;\left(\beta_{n,\mu}\right)^{1/2}\right)_{L_{p}}.$

Remark 2

The sequences $\beta_{n,\mu}$ and $\alpha_{n,\mu}$ in Theorem 5.2 satisfy $\beta_{n,\mu}={\cal{O}}(\tfrac{1}{n})$ and $\alpha_{n,\mu}={\cal{O}}(\tfrac{1}{n})$ as $n\rightarrow\infty$ .

Acknowledgements.

We are grateful to the referee for very helpful comments and suggestions.

References

(1) T. Acar, A. Aral, S. A. Mohiuddine, Approximation By Bivariate(p; q)-Bernstein-Kantorovichoperators, Iranian Journal of Science and Technology, Transactions A: Science, 1–8, (2016)
(2) F. Altomare, M. Campiti, Korovkin-type Approximation Theory and its Applications, Walter de Gruyter, Berlin-New York, 1994.
(3) F. Altomare V. Leonessa, On a sequence of positive linear operators associated with a continuous selection of Borel measures, Mediterr. J. Math. 3, 363–382, (2006)
(4) F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Raşa, A generalization of Kantorovich operators for convex compact subsets, Banach J. Math. Anal. 11, no. 3, 591–614, (2017).
(5) F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Raşa, Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators, J. Math. Anal. Appl. 458, No. 1, 153–173, (2018).
(6) A. Aral, V. Gupta, R. P. Agarwal, Applications of q-Calculus in Operator Theory, Springer, New York, 2013.
(7) A. Aral, D. Cárdenas-Morales and P. Garrancho, Bernstein-type operators that reproduce exponential functions, J. Math. Inequal. (Accepted.)
(8) R. A. Devore, “Degree of approximation,” in ApproximationTheory II, pp. 117–161, Academic Press, New York, NY, USA, 1976.
(9) Z. Ditzian and X. Zhou, Kantorovich-Bernstein polynomials. Constr. Approx. 6, 421–435, (1990).
(10) H. Gonska, P. Piţul, and I. Raşa On Peano’s form of the Taylor remainder, Voronovskaja’s theorem and the commutator of positive linear operators. In Proceedings of the International Conference on Numerical Analysis and Approximation Theory. Cluj Napoca, Romania, July 5–8, pp. 55–80, (2006).
(11) K. G. Ivanov, Approximation by Bernstein polynomials in $L_{p}$ metric. Constructive Function Theory ’84, Sofia, 421–429, (1984)
(12) L. V. Kantorovich, Sur certains d’eveloppements suivant les polynmes de la forme de S. Bernstein, I, II. C. R. Acad. URSS, 563-568 and 595–600, (1930).
(13) G.G. Lorentz, Approximation of functions, Chelsea Publ. Co., New York, 1986.
(14) S. Morigi and M. Neamtu, Some results for a class of generalized polynomials, Adv.Comput. Math. 12, 133–149, (2000).
(15) J. Peetre, A theory of interpolation of normed spaces, Lecture Notes, Brasilia, 1963 (Notas de Matematica, 39, (1968).
(16) R. Păltănea, A note on Bernstein Kantorovich operators, Bull. Univ. Transilvania of Braşov, Series III, Vol 6(55), No. 2, 27–32, (2013).
(17) P. C. Sikkema, Der Wert einiger Konstanten in der Theorie der Approximation mit Bernstein-Polynomen, Numer. Math. 3, 107–116, (1961).
(18) E. M. Stein, Singular Integrals, Princeton, New Jersey, NJ, USA, 1970.
(19) A. Zygmund, Trigonometric Series I, II. Cambridge University Press, Cambridge, UK, (1959).

2019

Unique continuation for the Helmholtz equation using stabilized finite element methods

August 22, 2019

Presentation by Alexandru Tamasan: A Fourier approach to the source reconstruction in a strongly scattering medium

July 16, 2019

Positive solutions for discontinuous systems via a multivalued vector version of Krasnosel’skii’s fixed point theorem in cones

April 28, 2022

	$\displaystyle\left\\|\widetilde{K}_{n}f-f\right\\|_{\infty}$	$\displaystyle\leq\sup_{x\in\left[0,1\right]}\left(a_{n+1}^{{}^{\prime}}\left(x\right)-1\right)e^{\mu}\left\\|B_{n}\left(\left(n+1\right)\delta_{n}^{\mu}\left(F\right);a_{n+1}\left(\cdot\right)\right)\right\\|_{\infty}$
		$\displaystyle\quad+\left(n+1\right)e^{\mu}\left\\|B_{n}\left(\delta_{n}^{\mu}\left(F\right);a_{n+1}\left(\cdot\right)\right)-\delta_{n}^{\mu}\left(F\right)\right\\|_{\infty}$
		$\displaystyle\quad+e^{\mu}\left\\|\left(n+1\right)\delta_{n}^{\mu}\left(F\right)-e^{-\mu\cdot}f\right\\|_{\infty}$
		$\displaystyle\leq b_{n}e^{\mu}\left\\|e^{-\mu\cdot}f\right\\|_{\infty}+\left(n+1\right)e^{\mu}\left\\|B_{n}\left(\delta_{n}^{\mu}\left(F\right);a_{n+1}\left(\cdot\right)\right)-B_{n}\left(\delta_{n}^{\mu}\left(F\right);\cdot\right)\right\\|_{\infty}$
		$\displaystyle\quad+\left(n+1\right)e^{\mu}\left\\|B_{n}\left(\delta_{n}^{\mu}\left(F\right);\cdot\right)-\delta_{n}^{\mu}\left(F\right)\right\\|_{\infty}$
		$\displaystyle\quad+e^{\mu}\left\\|\left(n+1\right)\delta_{n}^{\mu}\left(F\right)-e^{-\mu\cdot}f\right\\|_{\infty}$
		$\displaystyle:=b_{n}e^{\mu}\left\\|f\right\\|_{\infty}+I_{1}+I_{2}+I_{3}.$

	$\displaystyle\left\|B_{n}\left(\delta_{n}^{\mu}\left(F\right);a_{n+1}\left(x\right)\right)-B_{n}\left(\delta_{n}^{\mu}\left(F\right);x\right)\right\|$
	$\displaystyle\leq\omega\left(B_{n}\left(\delta_{n}^{\mu}\left(F\right)\right),\left\|a_{n+1}\left(x\right)-x\right\|\right)$
	$\displaystyle\leq 2\omega\left(\delta_{n}^{\mu}\left(F\right),\left\|a_{n+1}\left(x\right)-x\right\|\right)$
	$\displaystyle\leq 2\omega\left(\delta_{n}^{\mu}\left(F\right),\gamma_{n}\right),$		(29)

$\displaystyle\left\|R\left(f,x_{0};x\right)\right\|$	$\displaystyle=\frac{1}{2}\exp_{\mu,x_{0}}^{2}\left(x\right)\left\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\left(e^{\mu\xi_{x}}\right)-\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\left(e^{\mu x}\right)\right\|$
	$\displaystyle\leq\frac{1}{2}\exp_{\mu,x_{0}}^{2}\left(x\right)\omega\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\left\|\exp_{\mu,x}\left(\xi_{x}\right)\right\|\right)$
	$\displaystyle\leq\frac{1}{2}\exp_{\mu,x_{0}}^{2}\left(x\right)\omega\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\left\|\exp_{\mu,x}\left(x_{0}\right)\right\|\right),$	(32)

	$\displaystyle\left\|R\left(f,x_{0};x\right)\right\|$	$\displaystyle\leq\left\|R\left(f-g,x_{0};x\right)\right\|+\left\|R\left(g,x_{0};x\right)\right\|$
		$\displaystyle\leq\exp_{\mu,x_{0}}^{2}\left(x\right)\left\\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}-\left(g\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\right\\|_{\infty}$
		$\displaystyle\quad+\frac{1}{6}\left\|\exp_{\mu,x_{0}}^{3}\left(x\right)\right\|\left\\|\left(g\circ\log_{\mu}\right)^{{}^{{}^{{}^{\prime\prime\prime}}}}\right\\|_{\infty}.$

	$\displaystyle\left\|\widetilde{K}_{n}\left(R\left(f,x;t\right);x\right)\right\|=$
	$\displaystyle=\widetilde{K}_{n}\left(\exp_{\mu,x}^{2}\left(t\right)K\left(\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}};\frac{1}{6}\left\|\exp_{\mu,x}\left(t\right)\right\|,C,C^{1}\right);x\right)$
	$\displaystyle\leq\widetilde{K}_{n}\left(\exp_{\mu,x}^{2}\left(t\right)\left\{\left\\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}-\left(g\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\right\\|_{\infty}+\frac{\left\|\exp_{\mu,x}\left(t\right)\right\|}{6}\left\\|\left(g\circ\log_{\mu}\right)^{{}^{{}^{{}^{\prime\prime\prime}}}}\right\\|_{\infty}\right\};x\right)$
	$\displaystyle=\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)\left\\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}-\left(g\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\right\\|_{\infty}+\frac{1}{6}\widetilde{K}_{n}\left(\left\|\exp_{\mu,x}^{3}\right\|;x\right)\left\\|\left(g\circ\log_{\mu}\right)^{{}^{{}^{{}^{\prime\prime\prime}}}}\right\\|_{\infty}$
	$\displaystyle\leq\sqrt{\widetilde{K}_{n}\left(\exp_{\mu,x}^{4};x\right)}\sqrt{\widetilde{K}_{n}\left(e_{0};x\right)}\left\{\left\\|\left(f\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}-\left(g\circ\log_{\mu}\right)^{{}^{{}^{\prime\prime}}}\right\\|_{\infty}\right.$
	$\displaystyle\quad+\frac{1}{6}\frac{\sqrt{\widetilde{K}_{n}\left(\exp_{\mu,x}^{2};x\right)}}{\sqrt{\widetilde{K}_{n}\left(e_{0};x\right)}}\left\\|\left(g\circ\log_{\mu}\right)^{{}^{{}^{{}^{\prime\prime\prime}}}}\right\\|_{\infty}\Bigg\}.$

On approximation by some Bernstein–Kantorovich exponential-type polynomials

Abstract

Authors

Keywords

References

Paper coordinates

PDF

About this paper

Journal

Publisher Name

DOI

Print ISSN

Online ISSN

Google Scholar Profile

References

On approximation by some Bernstein-Kantorovich exponential-type polynomials

Abstract

1 Introduction

2 Preliminaries

Definition 1

Lemma 1

Theorem 2.1

Proof

Lemma 2

Proof

Lemma 3

Lemma 4

Proof

3 Convergence in C​[0,1]C\left[0,1\right]

Theorem 3.1

Proof

Theorem 3.2

Proof

Remark 1

4 Quantitative Voronovskaya theorem

Lemma 5

Proof

Theorem 4.1

Proof

Corollary 1

5 Convergence in Lp,μ​[0,1]L_{p,\mu}\left[0,1\right] and Lp​[0,1]L_{p}\left[0,1\right]

Lemma 6

Proof

Theorem 5.1

Proof

Lemma 7

Proof

Theorem 5.2

Proof

Remark 2

Acknowledgements.

References

Related Posts

3 Convergence in $C\left[0,1\right]$

5 Convergence in $L_{p,\mu}\left[0,1\right]$ and $L_{p}\left[0,1\right]$