Korovkin-Type Theorems in Weighted -Spaces via Summation Process

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Volume 2013, Article ID 534054,6pages http://dx.doi.org/10.1155/2013/534054

Research Article

Korovkin-Type Theorems in Weighted

𝐿

_𝑝

-Spaces via

Summation Process

Tuncer Acar

1

and Fadime Dirik

2

1_{Kırıkkale University, Faculty of Science and Arts, Department of Mathematics, Yahs¸ihan, Kırıkkale, Turkey} 2_{Sinop University, Faculty of Sciences and Arts, Department of Mathematics, 57000 Sinop, Turkey}

Correspondence should be addressed to Tuncer Acar; tunceracar@ymail.com Received 27 August 2013; Accepted 27 September 2013

Academic Editors: I. Beg, G. Dai, F. J. Garcia-Pacheco, H. Iiduka, and S. A. Mohiuddine

Copyright © 2013 T. Acar and F. Dirik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Korovkin-type theorem which is one of the fundamental methods in approximation theory to describe uniform convergence of

any sequence of positive linear operators is discussed on weighted𝐿_𝑝spaces,1 ≤ 𝑝 < ∞ for univariate and multivariate functions,

respectively. Furthermore, we obtain these types of approximation theorems by means ofA-summability which is a stronger

convergence method than ordinary convergence.

1. Introduction

The fundamental theorem of Korovkin [1] on approximation of continuous functions on a compact interval gives con-ditions in order to decide whether a sequence of positive linear operators converges to identity operator. This theorem has been extended in several directions. One of the most important papers on these extensions is [2] that where the author obtained Korovkin-type theorem on unbounded sets for the weighted continuous functions on semireal axis.

Korovkin-type theorems were also studied on𝐿_𝑝-spaces (see

[3,4]).

The extension of Korovkin’s theorem from compact

inter-vals to unbounded interinter-vals for functions that belong to𝐿_𝑝

-spaces was obtained by Gadjiev and Aral [5]. We recall some

notations presented in that paper. LetR denote the set of real

numbers. The function𝜔 is called a weight function if it is

positive continuous function on the whole real axis and, for a

fixed𝑝 ∈ [1, ∞), satisfying the condition

∫

R𝑡

2𝑝_{𝜔 (𝑡) 𝑑𝑡 < ∞.} ₍₁₎

Let 𝐿_𝑝,𝜔(R) (1 ≤ 𝑝 < ∞) denote the linear space of

measurable, 𝑝-absolutely integrable functions on R with

respect to weight function𝜔; that is,

𝐿_𝑝,𝜔(R)

={𝑓 : R 󳨀→ R; 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝,𝜔= (∫

R󵄨󵄨󵄨󵄨𝑓(𝑡)󵄨󵄨󵄨󵄨

𝑝_{𝜔 (𝑡) 𝑑𝑡)}1/𝑝_{< ∞}.}

(2) The analogues of (1) and (2) in multidimensional space

are given as follows. LetΩ be a positive continuous function

inR𝑛, satisfying the condition

∫

R𝑛|𝑡|

2𝑝_{Ω (𝑡) 𝑑𝑡 < ∞,} ₍₃₎

and for1 ≤ 𝑝 < ∞ one has

𝐿_𝑝,Ω(R𝑛)

={𝑓 : R𝑛󳨀→R; 󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝,Ω=(∫

R𝑛󵄨󵄨󵄨󵄨𝑓(𝑡)󵄨󵄨󵄨󵄨

𝑝_{Ω (𝑡) 𝑑𝑡)}1/𝑝_<∞}.

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The authors obtained Korovkin-type theorems for the

functions in 𝐿_𝑝,𝜔(R) and also in 𝐿_𝑝,Ω(R𝑛). The

aforemen-tioned results are the extensions of Korovkin’s theorem on unbounded sets and more general functions spaces by ordi-nary convergence.

On the other hand, most of the classical operators tend to converge to the value of the function being approximated. At the points of discontinuity, they often converge to the average of the left and right limits of the function. However, there are exceptions which do not converge at points of discontinuity (see [6]). In this case matrix summability methods of Ces´aro type are strong enough to correct the lack of convergence [7].

LetA := {𝐴𝑛} = (𝑎_𝑘𝑗(𝑛)) be a sequence of infinite matrices

with non-negative real entries. For a sequence𝑥 = (𝑥_𝑗), the

double sequence A𝑥 := {(𝐴𝑥)𝑛_𝑘 : 𝑘, 𝑛 ∈ N} , (5) defined by (𝐴𝑥)𝑛_𝑘 :=∑∞ 𝑗=1 𝑎_𝑘𝑗(𝑛)𝑥_𝑗, (6)

is calledA-transform of 𝑥 whenever the series that converges

for all𝑘, 𝑛, and 𝑥 is said to be A-summable to 𝑙 if

lim 𝑘 → ∞ ∞ ∑ 𝑗=1 𝑎(𝑛)_𝑘𝑗𝑥_𝑗= 𝑙, (7)

uniformly in𝑛 ([8,9]). If𝐴𝑛 = 𝐴 for some matrices 𝐴, then

A-summability is the ordinary matrix summability by 𝐴, and

if𝑎_𝑘𝑗(𝑛) = 1/𝑘, for 𝑛 ≤ 𝑗 ≤ 𝑘 + 𝑛 (𝑛 = 1, 2, 3, . . .), and 𝑎_𝑘𝑗(𝑛) =

0 otherwise, then A-summability reduces to almost

con-vergence [10]. Replacing the ordinary concon-vergence by

A-summability some approximation results have been studied

in [11–13] and in the special cases [14,15]. Also, Korovkin-type

theorems in weighted space via A-summability have been

studied in [16,17].

Our purpose in the present paper is to obtain

Korovkin-type theorems on weighted 𝐿_𝑝 spaces in univariate and

multivariate case viaA-summation process. More precisely,

a sequence{𝐿_𝑗} of positive linear operators from 𝐿_𝑝,𝜔into

𝐿_𝑝,𝜔is called anA-summation process on 𝐿_𝑝,𝜔if(𝐿_𝑗(𝑓)) is

A-summable to 𝑓 for every 𝑓 ∈ 𝐿_𝑝,𝜔; that is,

lim 𝑘 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩󵄩 󵄩󵄩󵄩 ∞ ∑ 𝑗=1 𝑎_𝑘𝑗(𝑛)𝐿_𝑗(𝑓) − 𝑓󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 󵄩󵄩󵄩𝑝,𝜔= 0, uniformly in 𝑛, (8)

where it is assumed that the series in (8) converges for each 𝑘, 𝑛, and 𝑓. Considering this fact we extend (8) to space of sequences of linear positive operators to approximate the

functions that belong to𝐿_𝑝,𝜔spaces via matrix summability

method.

2. Main Result

Throughout this section we will use the following notations:

𝐴(𝑛)_{(𝑓; 𝑥) is the double sequence:}

𝐴(𝑛)_𝑘 (𝑓; 𝑥) =∑∞

𝑗=1

𝑎_𝑘𝑗(𝑛)𝐿_𝑗(𝑓 (𝑡) ; 𝑥) , 𝑛 = 1, 2, . . . , (9)

and minimum and maximum values of the weight function

𝜔 on finite intervals will be denoted by 𝜔_min and 𝜔_max,

respectively.

Now we present the following main result.

Theorem 1. Let A = {𝐴𝑛_{} be a sequence of infinite matrices}

with nonnegative real entries and let {𝐿_𝑗} be a sequence of

positive linear operators from𝐿_𝑝,𝜔into𝐿_𝑝,𝜔. Assume that

sup 𝑛,𝑘󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 󵄩󵄩󵄩󵄩_{󵄩𝐿𝑝,𝜔}→ 𝐿𝑝,𝜔 < ∞. (10) If lim 𝑘 sup𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑡𝑖; 𝑥) − 𝑥𝑖󵄩󵄩󵄩󵄩󵄩𝑝,𝜔= 0, 𝑖 = 0, 1, 2, (11)

then for any function𝑓 ∈ 𝐿_𝑝,𝜔(R), one has

lim

𝑘 sup𝑛 󵄩󵄩󵄩󵄩󵄩𝐴

(𝑛)

𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔= 0. (12)

Proof. Let𝜒₁𝐴(𝑡) be the characteristic function of the interval

[−𝐴, 𝐴] and 𝜒𝐴

2(𝑡) = 1 − 𝜒1𝐴(𝑡) for any 𝐴 ≥ 0. We can choose

a sufficient large𝐴 such that for every 𝜀 > 0

󵄩󵄩󵄩󵄩

󵄩𝑓𝜒2𝐴󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔}< 𝜀. (13)

Using the assumption of the convergence of the series (9)

for each𝑘, 𝑛, and 𝑓 and the linearity of the operators 𝐿_𝑗, we

get sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔 = sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝜒𝐴1 + 𝜒2𝐴) 𝑓 − (𝜒𝐴1 + 𝜒2𝐴) 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔 ≤ sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝜒𝐴1𝑓) − 𝜒1𝐴𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔 + sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝜒2𝐴𝑓) − 𝜒𝐴2𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔 = sup 𝑛 𝐼 󸀠 𝑛+ sup_𝑛 𝐼𝑛󸀠󸀠. (14)

By condition (10), there exists a constant𝐾 > 0 such that

sup

𝑛,𝑘󵄩󵄩󵄩󵄩󵄩𝐴

(𝑛)

𝑘 󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔}≤ 𝐾. (15)

Hence, from (13), we compute sup 𝑛 𝐼 󸀠󸀠 𝑛 ≤ sup_𝑛 󵄩󵄩󵄩󵄩_󵄩𝐴(𝑛)𝑘 𝜒2𝐴𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔+ 󵄩󵄩󵄩󵄩󵄩𝜒𝐴2𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔 ≤ (𝐾 + 1)󵄩󵄩󵄩󵄩_󵄩𝜒₂𝐴_{𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔} < (𝐾 + 1) 𝜀. (16)

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For every function𝑓 ∈ 𝐿_𝑝,𝜔(R) the inequality 󵄩󵄩󵄩󵄩

󵄩𝜒𝐴2𝑓󵄩󵄩󵄩󵄩󵄩𝑝≤ 𝜔min−1/𝑝󵄩󵄩󵄩󵄩𝑓󵄩󵄩󵄩󵄩𝑝,𝜔 (17)

implies that𝐿_𝑝,𝜔(−𝐴, 𝐴) ⊂ 𝐿_𝑝(−𝐴, 𝐴). Since the space of

con-tinuous functions is dense in𝐿_𝑝(−𝐴, 𝐴), given 𝑓 ∈ 𝐿_𝑝,𝜔(R),

for each 𝜀󸀠 > 0, there exists a continuous function 𝜑 on

[−𝐴, 𝐴] satisfying the condition 𝜑(𝑥) = 0 for |𝑥| > 𝐴 such that 󵄩󵄩󵄩󵄩 󵄩(𝑓 − 𝜑) 𝜒𝐴1󵄩󵄩󵄩󵄩_󵄩𝑝< 𝜀 󸀠 (𝐾 + 1) 𝜔max1/𝑝 . (18)

Using the inequalities (15) and (18), we get sup 𝑛 𝐼 󸀠 𝑛= sup_𝑛 󵄩󵄩󵄩󵄩_󵄩𝐴(𝑛)𝑘 (𝜒𝐴1𝑓) − 𝜒1𝐴𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔 ≤ sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑓 − 𝜑) 𝜒𝐴1󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} + sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝜑𝜒1𝐴) − 𝜑𝜒1𝐴󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔}+ 󵄩󵄩󵄩󵄩󵄩(𝑓 − 𝜑)𝜒𝐴1󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} ≤ sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝜑𝜒1𝐴) − 𝜑𝜒𝐴1󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔}+ 𝜀󸀠. (19)

On the other hand, since𝜒₂𝐴1𝜒𝐴₁𝜑 = 0 for some 𝐴₁ > 𝐴,

we get the inequality sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝜑𝜒1𝐴) − 𝜑𝜒1𝐴󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} = sup 𝑛 󵄩󵄩󵄩󵄩󵄩(𝜒 𝐴1 1 + 𝜒2𝐴1) 𝐴(𝑛)𝑘 (𝜑𝜒1𝐴) − (𝜒𝐴11 + 𝜒2𝐴1) 𝜑𝜒1𝐴󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} ≤ sup 𝑛 󵄩󵄩󵄩󵄩󵄩[𝐴 (𝑛) 𝑘 (𝜑𝜒𝐴1) − 𝜑𝜒1𝐴] 𝜒𝐴11 󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} + sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝜒 𝐴1 2 𝐴(𝑛)𝑘 (𝜑𝜒1𝐴)󵄩󵄩󵄩󵄩󵄩𝑝,𝜔. (20)

Now, by supposing that𝑀_𝜑= max_𝑡∈R|𝜑(𝑡)|𝜒₁𝐴(𝑡), we get

sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝜒 𝐴1 2 𝐴(𝑛)𝑘 (𝜑𝜒1𝐴)󵄩󵄩󵄩󵄩󵄩𝑝,𝜔 = sup 𝑛 (∫|𝑡|>𝐴1󵄨󵄨󵄨󵄨󵄨𝐴 (𝑛) 𝑘 (𝜑𝜒𝐴1; 𝑡)󵄨󵄨󵄨󵄨󵄨 𝑝 𝜔 (𝑡) 𝑑𝑡)1/𝑝 ≤ 𝑀_𝜑sup 𝑛 (∫|𝑡|>𝐴1󵄨󵄨󵄨󵄨󵄨𝐴 (𝑛) 𝑘 (1; 𝑡) − 1󵄨󵄨󵄨󵄨_󵄨 𝑝 𝜔 (𝑡) 𝑑𝑡)1/𝑝 + 𝑀_𝜑(∫ R𝜒 𝐴1 2 𝜔 (𝑡) 𝑑𝑡) 1/𝑝 . (21)

Since𝜔 ∈ 𝐿₁(R), we can choose the number 𝐴₁such that

(∫ R𝜒 𝐴1 2 𝜔 (𝑡) 𝑑𝑡) 1/𝑝 < _𝑀𝜀󸀠 𝜑. (22)

Using this inequality, we have sup

𝑛 󵄩󵄩󵄩󵄩󵄩𝜒

𝐴1

2 𝐴(𝑛)𝑘 (𝜑𝜒1𝐴)󵄩󵄩󵄩󵄩󵄩𝑝,𝜔≤ 𝑀𝜑sup_𝑛 󵄩󵄩󵄩󵄩_󵄩𝐴(𝑛)𝑘 (1; 𝑥) − 1󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔}+ 𝜀󸀠.

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As a corollary, we get the following inequality for sup_𝑛𝐼_𝑛󸀠:

sup 𝑛 𝐼 󸀠 𝑛≤ 2𝜀󸀠+ 𝑀𝜑sup_𝑛 󵄩󵄩󵄩󵄩_󵄩𝐴(𝑛)𝑘 (1; 𝑥) − 1󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} + sup 𝑛 󵄩󵄩󵄩󵄩󵄩[𝐴 (𝑛) 𝑘 (𝜑𝜒1𝐴) − 𝜑𝜒1𝐴] 𝜒𝐴11󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔}. (24)

Since𝜑𝜒𝐴₁ is a continuous function on [−𝐴, 𝐴], for any

given𝜀󸀠> 0, there exists a 𝛿 > 0 such that

󵄨󵄨󵄨󵄨

󵄨𝜑 (𝑡) 𝜒𝐴1 (𝑡) − 𝜑 (𝑥) 𝜒1𝐴(𝑥)󵄨󵄨󵄨󵄨_{󵄨 < 𝜀}󸀠+ 2𝑀𝜑(𝑡 − 𝑥)

2

𝛿2 . (25)

Furthermore, this inequality also holds in the case that𝑡 ∈

[−𝐴, 𝐴] and that 𝑥 ∈ [−𝐴₁, −𝐴) ∪ (𝐴, 𝐴₁] since 𝜑(𝑥)𝜒𝐴

1(𝑥) =

0 and 𝜑(𝑡)𝜒₁𝐴(𝑡) are continuous. So, we have

sup 𝑛 󵄩󵄩󵄩󵄩󵄩[𝐴 (𝑛) 𝑘 (𝜑𝜒1𝐴) − 𝜑𝜒𝐴1] 𝜒1𝐴1󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} ≤ sup 𝑛 󵄩󵄩󵄩󵄩󵄩[𝐴 (𝑛) 𝑘 (󵄨󵄨󵄨󵄨󵄨𝜑(𝑡)𝜒𝐴1 (𝑡) − 𝜑 (𝑥) 𝜒1𝐴(𝑥)󵄨󵄨󵄨󵄨_{󵄨 ; 𝑥)] 𝜒}1𝐴1(𝑥)󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} + sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝜑 (𝑥) 𝜒 𝐴 1 (𝑥) (𝐴(𝑛)𝑘 (1; 𝑥) − 1)󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} ≤ (𝜀󸀠+2𝑀𝜑 𝛿2 𝐴2+ 𝑀𝜑) sup_𝑛 󵄩󵄩󵄩󵄩_󵄩𝐴(𝑛)𝑘 (1; 𝑥) − 1󵄩󵄩󵄩󵄩󵄩𝑝,𝜔+ 𝜀󸀠 +4𝑀𝜑 𝛿2 𝐴sup_𝑛 󵄩󵄩󵄩󵄩󵄩𝐴(𝑛)𝑘 (𝑡; 𝑥) − 𝑥󵄩󵄩󵄩󵄩󵄩𝑝,𝜔 +2𝑀𝜑 𝛿2 sup_𝑛 󵄩󵄩󵄩󵄩󵄩𝐴(𝑛)𝑘 (𝑡2; 𝑥) − 𝑥2󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔}. (26) Using (24) and (26), we can write

sup 𝑛 𝐼 󸀠 𝑛 ≤ 3𝜀󸀠+ (𝜀󸀠+2𝑀_𝛿₂𝜑𝐴2+ 2𝑀_𝜑) sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (1; 𝑥) − 1󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} +4𝑀𝜑 𝛿2 𝐴sup_𝑛 󵄩󵄩󵄩󵄩󵄩𝐴(𝑛)𝑘 (𝑡; 𝑥) − 𝑥󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} +2𝑀𝜑 𝛿2 sup_𝑛 󵄩󵄩󵄩󵄩󵄩𝐴(𝑛)𝑘 (𝑡2; 𝑥) − 𝑥2󵄩󵄩󵄩󵄩󵄩𝑝,𝜔. (27)

(4)

Then we obtain the following equality for (14): sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔 ≤ 3𝜀󸀠+ (𝐾 + 1) 𝜀 + 𝐶 {sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (1; 𝑥) − 1󵄩󵄩󵄩󵄩󵄩𝑝,𝜔 + sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑡; 𝑥) − 𝑥󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} + sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑡2; 𝑥) − 𝑥2󵄩󵄩󵄩󵄩󵄩𝑝,𝜔} , (28) where 𝐶 := max{𝜀󸀠 + (2𝑀_𝜑/𝛿2)𝐴2 + 2𝑀_𝜑, (4𝑀_𝜑/𝛿2)𝐴,

(2𝑀_𝜑/𝛿2_{)}. By the hypothesis of theorem and arbitrarity of 𝜀}

and𝜀󸀠, sup_𝑛‖𝐴(𝑛)_𝑘 𝑓 − 𝑓‖_𝑝,𝜔 → 0 as 𝑘 → ∞ which is desired

result.

Now we give an example of a sequence of positive linear

operators which satisfies the conditions of Theorem 1 in

weighted space𝐿_𝑝,𝜔(R).

Example 2. We choose𝜔(𝑥) = 𝑒−𝑥. Note that this selection of

𝜔 satisfies condition (1). Also note that for1 ≤ 𝑝 < ∞

𝐿𝑝,𝜔(R) = {𝑓 : R 󳨀→ R : 𝑒−𝑥𝑓 (𝑥) ∈ 𝐿𝑝(R)} . (29)

Also𝐴𝑛= 𝐶 for each 𝑛 where 𝐶 is the Ces´aro matrix; that

is, 𝑐_𝑘𝑗={_{ { 1 𝑘, 1 ≤ 𝑗 ≤ 𝑘, 0, otherwise. (30)

The Kantorovich variant of the Sz´asz-Mirakyan operators

[18] by replacing𝑓(𝑠𝑏_𝑘/𝑘) with an integral mean of 𝑓(𝑥) over

the interval[(𝑠 + 1)𝑏_𝑘/𝑘, 𝑠𝑏_𝑘/𝑘] is as follows:

𝑆_𝑘(𝑓; 𝑥) := _𝑏𝑘 𝑘 ∞ ∑ 𝑠=0𝑃𝑘,𝑠(𝑥) ∫ (𝑠+1)𝑏𝑘/𝑘 𝑠𝑏𝑘/k 𝑓 (𝑡) 𝑑𝑡, 𝑘 ∈ N, 𝑥 ∈ [0, 𝑏_𝑘) , (31)

where(𝑏_𝑘) is a sequence of positive real numbers satisfying

the condition lim 𝑘 → ∞ 𝑏_𝑘 𝑘 = 0, 𝑘 → ∞lim𝑏𝑘= ∞, 𝑃_𝑘,𝑠(𝑥) := 𝑒−𝑘𝑥/𝑏𝑘(𝑘𝑥)_𝑠!𝑏_𝑠𝑠 𝑘 , 𝑠 = 0, 1, 2, . . . . (32) It is known that 𝑆𝑘(1; 𝑥) = 1, 𝑆𝑘(𝑡; 𝑥) = 𝑥 +_2𝑘𝑏𝑘, 𝑆_𝑘(𝑡2; 𝑥) = 𝑥2+2𝑏𝑘 𝑘 𝑥 + 𝑏2 𝑘 3𝑘2. (33)

Furthermore by simple calculations, we obtain sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (1; 𝑥) − 1󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔}= 0, sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑡; 𝑥) − 𝑥󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔}= _2𝑘1 ‖1‖𝑝,𝜔 𝑘 ∑ 𝑗=1 𝑏_𝑗 𝑗, sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑡2; 𝑥) − 𝑥2󵄩󵄩󵄩󵄩_{󵄩𝑝,𝜔} ≤ 2_𝑘‖𝑥‖_𝑝,𝜔∑𝑘 𝑗=1 𝑏_𝑗 𝑗 + 1 3𝑘‖1‖𝑝,𝜔 𝑘 ∑ 𝑗=1 𝑏2 𝑗 𝑗2. (34) Also, sup 𝑛,𝑘󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 󵄩󵄩󵄩󵄩󵄩𝐿𝑝,𝜔→ 𝐿𝑝,𝜔 = sup 𝑛,𝑘_‖𝑓sup_‖𝑝,𝜔=1󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑓; 𝑥)󵄩󵄩󵄩󵄩󵄩𝑝,𝜔< ∞. (35)

Hence, conditions (10), (11) are provided which means

that for any function𝑓 ∈ 𝐿_𝑝,𝜔(R), we have

lim

(𝑛)

𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,𝜔= 0. (36)

Also, analogue of Theorem 1 for the space of function

of several variables can be obtained. Now, we establish this theorem. For the sake of convenient notation, we present our

second results on𝐿_𝑝,Ω(R𝑚), 𝑚 ∈ N, instead of 𝐿_𝑝,Ω(R𝑛) to

avoid any confusion about the indices ofA = {𝐴𝑛}.

Theorem 3. Let A = {𝐴𝑛_{} be a sequence of infinite matrices}

with nonnegative real entries and let{𝐿_𝑗} be a sequence of

pos-itive linear operators from𝐿_𝑝,Ω(R𝑚) into 𝐿_𝑝,Ω(R𝑚). Assume

that sup 𝑛,𝑘󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 󵄩󵄩󵄩󵄩󵄩𝐿𝑝,Ω→ 𝐿𝑝,Ω < ∞. (37) If lim 𝑘 sup𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (1; 𝑥) − 1󵄩󵄩󵄩󵄩󵄩𝑝,Ω= 0, lim 𝑘 sup𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑡𝑖; 𝑥) − 𝑥𝑖󵄩󵄩󵄩󵄩_󵄩𝑝,Ω= 0, 𝑖 = 1, 2, . . . , 𝑚, lim 𝑘 sup𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (|𝑡|2; 𝑥) − |𝑥|2󵄩󵄩󵄩󵄩_󵄩𝑝,Ω= 0, (38)

then for any function𝑓 ∈ 𝐿_𝑝,Ω(R𝑚), one has

lim

(𝑛)

𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,Ω= 0. (39)

Proof. Considering the characteristic function𝜒𝐴₁ of the ball

|𝑥| ≤ 𝐴 and 𝜒𝐴

2(𝑡) = 1 − 𝜒1𝐴(𝑡), it is possible to choose a

sufficient large𝐴 such that

󵄩󵄩󵄩󵄩

(5)

On the other hand, by condition (37) there exists a

pos-itive constant𝐾 such that ‖𝐴(𝑛)_𝑘 ‖_𝑝,Ω ≤ 𝐾, and so, for given

𝜀󸀠 _{> 0, there exists a continuous function 𝜃 on |𝑥| ≤ 𝐴}

satisfying the condition𝜃(𝑥) = 0 for |𝑥| > 𝐴 such that

󵄩󵄩󵄩󵄩

󵄩(𝑓 − 𝜃) 𝜒1𝐴󵄩󵄩󵄩󵄩_󵄩𝑝,Ω< 𝜀

󸀠

(𝐾 + 1) (max_{|𝑡|≤𝐴}Ω (𝑡))1/𝑝. (41)

Keeping in mind the fact that the series (9) is a

conver-gence for each𝑘, 𝑛, and 𝑓, and using the linearity of the

oper-ators𝐿_𝑗, which means the linearity of𝐴(𝑛)_𝑘 , we get

sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,Ω≤ sup_𝑛 󵄩󵄩󵄩󵄩󵄩𝐴(𝑛)𝑘 (𝜒1𝐴𝜃) − 𝜒1𝐴𝜃󵄩󵄩󵄩󵄩󵄩𝑝,Ω + (𝐾 + 1) 𝜀 + 𝜀󸀠. (42)

Let𝐴₁> 𝐴, so we also have

sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝜒1𝐴𝜃) − 𝜒𝐴1𝜃󵄩󵄩󵄩󵄩󵄩𝑝,Ω ≤ sup 𝑛 󵄩󵄩󵄩󵄩󵄩[𝐴 (𝑛) 𝑘 (𝜒𝐴1𝜃) − 𝜒1𝐴𝜃] 𝜒1𝐴1󵄩󵄩󵄩󵄩_󵄩𝑝,Ω + 𝑀_𝜃sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (1) − 1󵄩󵄩󵄩󵄩_󵄩𝑝,Ω + 𝑀𝜃󵄩󵄩󵄩󵄩_󵄩𝜒𝐴1₂ 󵄩󵄩󵄩󵄩_󵄩𝑝,Ω, (43)

where 𝑀_𝜃 := max_𝑡∈R𝑚|𝜃(𝑡)|𝜒𝐴₁(𝑡). Furthermore, we can

choose𝐴₁such that‖𝜒𝐴1₂ ‖_𝑝,Ω < 𝜀󸀠/𝑀_𝜃, and for sufficiently

large 𝑘, we estimate sup_𝑛‖𝐴(𝑛)_𝑘 (1) − 1‖_𝑝,Ω < 𝜀󸀠/𝑀_𝜃. Using

these estimations in (42), we obtain sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,Ω≤ sup_𝑛 󵄩󵄩󵄩󵄩_󵄩[𝐴(𝑛)𝑘 (𝜒1𝐴𝜃) − 𝜒𝐴1𝜃] 𝜒𝐴11 󵄩󵄩󵄩󵄩_󵄩𝑝,Ω + (𝐾 + 1) 𝜀 + 3𝜀󸀠. (44) Since 󵄨󵄨󵄨󵄨 󵄨𝜒1𝐴(𝑡) 𝜃 (𝑡) − 𝜒1𝐴(𝑥) 𝜃 (𝑥)󵄨󵄨󵄨󵄨_{󵄨 < 𝜀}󸀠+ 2𝑀𝜃|𝑡 − 𝑥| 2 𝛿2 , (45) we can write sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,Ω ≤ (𝐾 + 1) 𝜀 + 4𝜀󸀠𝐾‖Ω‖1/𝑃₁ + 2𝑀_𝜃(1 + 𝐴)_𝛿₂ 2{sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (|𝑡|2; 𝑥) − |𝑥|2󵄩󵄩󵄩󵄩_󵄩𝑝,Ω +∑𝑚 𝑙=0 sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑡𝑖; 𝑥) − 𝑥𝑖󵄩󵄩󵄩󵄩_󵄩𝑝,Ω + sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (1) − 1󵄩󵄩󵄩󵄩_󵄩𝑝,Ω} . (46)

Using the conditions of theorem, we have sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,Ω ≤ (𝐾 + 1) 𝜀 + 4𝜀󸀠𝐾‖Ω‖1/𝑃1 + 2𝑀𝜃(1 + 𝐴) 2 𝛿2 × [ 𝛿2𝜀∗1 2𝑀_𝜃(1 + 𝐴)2 + 𝑚 𝛿2_𝜀∗ 2 2𝑀_𝜃(1 + 𝐴)2 + 𝛿2_𝜀∗ 3 2𝑀_𝜃(1 + 𝐴)2] = (𝐾 + 1) 𝜀 + 4𝜀󸀠𝐾‖Ω‖1/𝑃1 + 𝜀1∗+ 𝑚𝜀∗2 + 𝜀3∗, (47) which means that

lim

(𝑛)

𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,Ω= 0. (48)

Now we give the following example.

Example 4. We chooseΩ(𝑥, 𝑦) = 𝑒−𝑥−𝑦. Note that this

selec-tion ofΩ satisfies condition (3). Also note that, for1 ≤ 𝑝 <

∞,

𝐿_𝑝,Ω(R2) = {𝑓 : R2󳨀→ R : Ω (𝑥, 𝑦) 𝑓 (𝑥) ∈ 𝐿𝑝(R2)} .

(49)

Also𝐴𝑛= 𝐶 for each 𝑛 where 𝐶 is the Ces´aro matrix, that

is; 𝑐_𝑘𝑗={_{ { 1 𝑘, 1 ≤ 𝑗 ≤ 𝑘, 0 otherwise. (50)

The Kantorovich variant of the Sz´asz-Mirakyan operators

[18] by replacing 𝑓(𝑡𝑏_𝑘/𝑘, 𝑠𝑏_𝑘/𝑘) with an integral mean of

𝑓(𝑥, 𝑦) over the interval [(𝑡 + 1)𝑏_𝑘/𝑘, 𝑡𝑏_𝑘/𝑘] × [(𝑠 + 1)𝑏_𝑘/

𝑘, 𝑠𝑏_𝑘/𝑘] is as follows: 𝑆_𝑘(𝑓; 𝑥, 𝑦) := 𝑘2 𝑏2 𝑘 ∞ ∑ 𝑡=0 ∞ ∑ 𝑠=0 𝑃_{𝑘,𝑡,𝑠}(𝑥, 𝑦) × ∫(𝑡+1)𝑏𝑘/𝑘 𝑡𝑏𝑘/𝑘 ∫ (𝑠+1)𝑏𝑘/𝑘 𝑠𝑏𝑘/𝑘 𝑓 (𝑢, V) 𝑑𝑢 𝑑V, 𝑘 ∈ N, 𝑥, 𝑦 ∈ [0, 𝑏_𝑘) , (51)

where(𝑏_𝑘) is a sequence of positive real numbers satisfying

the condition lim 𝑘 → ∞ 𝑏_𝑘 𝑘 = 0, 𝑘 → ∞lim𝑏𝑘= ∞, 𝑃_{𝑘,𝑡,𝑠}(𝑥, 𝑦) := 𝑒−(𝑘(𝑥+𝑦)/𝑏𝑘)(𝑘𝑥) 𝑡_(𝑘𝑦)𝑠 𝑡!𝑠!𝑏𝑡+𝑠 𝑘 , 𝑡, 𝑠 = 0, 1, 2, . . . . (52)

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It is known that 𝑆𝑘(1; 𝑥, 𝑦) = 1, 𝑆_𝑘(𝑢; 𝑥, 𝑦) = 𝑥 +_2𝑘𝑏𝑘, 𝑆_𝑘(V; 𝑥, 𝑦) = 𝑦 +_2𝑘𝑏𝑘, 𝑆_𝑘(𝑢2+ V2; 𝑥, 𝑦) = 𝑥2+ 𝑦2+2𝑏𝑘 𝑘 (𝑥 + 𝑦) + 2𝑏2 𝑘 3𝑘2. (53) Furthermore we obtain sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (1; 𝑥, 𝑦) − 1󵄩󵄩󵄩󵄩󵄩𝑝,Ω= 0, sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑢; 𝑥, 𝑦) − 𝑥󵄩󵄩󵄩󵄩󵄩𝑝,Ω= _2𝑘1 ‖1‖𝑝,Ω 𝑘 ∑ 𝑗=1 𝑏_𝑗 𝑗, sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (V; 𝑥, 𝑦) − 𝑦󵄩󵄩󵄩󵄩󵄩𝑝,Ω= _2𝑘1 ‖1‖𝑝,Ω 𝑘 ∑ 𝑗=1 𝑏_𝑗 𝑗, sup 𝑛 󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑢2+ V2; 𝑥, 𝑦) − (𝑥2+ 𝑦2)󵄩󵄩󵄩󵄩󵄩𝑝,Ω ≤ 2 𝑘󵄩󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩󵄩𝑝,Ω 𝑘 ∑ 𝑗=1 𝑏_𝑗 𝑗 + 2 3𝑘‖1‖𝑝,Ω 𝑘 ∑ 𝑗=1 𝑏2 𝑗 𝑗2. (54) Also, sup 𝑛,𝑘󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 󵄩󵄩󵄩󵄩󵄩𝐿𝑝,Ω→ 𝐿𝑝,Ω = sup 𝑛,𝑘_‖𝑓sup‖𝑝,Ω=1󵄩󵄩󵄩󵄩󵄩𝐴 (𝑛) 𝑘 (𝑓; 𝑥, 𝑦)󵄩󵄩󵄩󵄩󵄩𝑝,Ω< ∞. (55)

Hence, conditions (10) and (11) are provided which means

that for any function𝑓 ∈ 𝐿_𝑝,Ω(R2), we have

lim

(𝑛)

𝑘 𝑓 − 𝑓󵄩󵄩󵄩󵄩󵄩𝑝,Ω= 0. (56)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are thankful to referee(s) for making valuable suggestions leading to the better presentation of the paper.

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