Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 514174, 7 pages
http://dx.doi.org/10.1155/2013/514174
Research Article
A Generalization of Lacunary Equistatistical Convergence of
Positive Linear Operators
Yusuf Kaya and Nazmiye Gönül
Department of Mathematics, Faculty of Arts and Sciences, Bulent Ecevit University, 67100 Zonguldak, Turkey
Correspondence should be addressed to Nazmiye Gönül; nazmiyegonul81@hotmail.com
Received 25 January 2013; Revised 17 April 2013; Accepted 18 April 2013
Academic Editor: Adem Kılıçman
Copyright © 2013 Y. Kaya and N. Gönül. 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.
In this paper we consider some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence. In
particular we study lacunary equi-statistical convergence of approximating operators on 𝐻𝑤2 spaces, the spaces of all real valued
continuous functions 𝑓 dened on 𝐾 = [0, ∞)𝑚 and satisfying some special conditions.
1. Introduction
Approximation theory has important applications in the
theory of polynomial approximation, in various areas of
functional analysis, numerical solutions of integral and
differential equations [1–6]. In recent years, with the help
of the concept of statistical convergence, various statistical
approximation results have been proved [7]. In the usual
sense, every convergent sequence is statistically convergent,
but its converse is not always true. And, statistical convergent
sequences do not need to be bounded.
Recently, Aktuğlu and Gezer [8] generalized the idea of
statistical convergence to lacunary equi-statistical convergences. In this paper, we first study some Korovkin type
approximation theorems via lacunary equi- statistical convergence in 𝐻𝑤2 spaces. Then using the modulus of continuity,
we study rates of lacunary equi-statistically convergence in
𝐻𝑤2 .
We recall here the concepts of equi-statistical convergence and lacunary equi-statistical convergence.
Let 𝑓 and 𝑓𝑟 belong to 𝐶(𝑋), which is the space of
all continuous real valued functions on a compact subset
𝑋 of the real numbers. {𝑓𝑟 } is said to be equi-statistically
convergent to 𝑓 on 𝑋 and denoted by 𝑓𝑟 󴀀󴀤 𝑓 (equistat) if
for every 𝜀 > 0, the sequence of real valued functions
𝑝𝑟,𝜀 (𝑥) :=
1 󵄨󵄨
󵄨
󵄨{𝑚 ≤ 𝑟 : 𝑓𝑚 (𝑥) − 𝑓 (𝑥) ≥ 𝜀}󵄨󵄨󵄨
𝑟󵄨
(1)
converges uniformly to the zero function on 𝑋, which means
that
󵄩
󵄩
lim 󵄩󵄩𝑝 (⋅)󵄩󵄩󵄩𝐶(𝑋) = 0.
(2)
𝑟 → ∞󵄩 𝑟,𝜀
A lacunary sequence 𝜃 = {𝑘𝑟 } is an integer sequence such that
𝑘0 = 0,
ℎ𝑟 = 𝑘𝑟 − 𝑘𝑟−1 󳨀→ ∞
as 𝑟 󳨀→ ∞.
(3)
In this paper the intervals determined by 𝜃 will be denoted by
𝐼𝑟 = (𝑘𝑟−1 , 𝑘𝑟 ], and the ratio 𝑘𝑟 /𝑘𝑟−1 will be abbreviated by 𝑞𝑟 .
Let 𝜃 be a lacunary sequence then {𝑓𝑟 }𝑟∈N is said to be
lacunary equi-statistically convergent to 𝑓 on 𝑋 and denoted
by 𝑓𝑟 󴀀󴀤 𝑓 (𝜃-equistat) if for every 𝜀 > 0, the sequence of
real valued functions {𝑠𝑟,𝜀 }𝑛∈N defined by
𝑠𝑟,𝜀 (𝑥) :=
1
ℎ𝑟
󵄨󵄨
󵄨
󵄨󵄨{𝑚 ∈ 𝐼𝑟 : 𝑓𝑚 (𝑥) − 𝑓 (𝑥) ≥ 𝜀}󵄨󵄨󵄨
(4)
uniformly converges to zero function on 𝑋, which means that
󵄩
󵄩
lim 󵄩󵄩𝑠 (⋅)󵄩󵄩󵄩𝐶(𝑋) = 0.
(5)
𝑛 → ∞󵄩 𝑟,𝜀
A Korovkin type approximation theorem by means of
lacunary equi-statistical convergence was given in [8]. We can
state this theorem now. An operator 𝐿 defined on a linear
space of functions 𝑌 is called linear if 𝐿(𝛼𝑓 + 𝛽𝑔, 𝑥) =
𝛼𝐿(𝑓, 𝑥) + 𝛽𝐿(𝑔, 𝑥), for all 𝑓, 𝑔 ∈ 𝑌, 𝛼, 𝛽 ∈ R and is called
positive, if 𝐿(𝑓, 𝑥) ≥ 0, for all 𝑓 ∈ 𝑌, 𝑓 ≥ 0. Let 𝑋 be a compact
subset of R, and let 𝐶(𝑋) be the space of all continuous real
valued functions on 𝑋.
2
Abstract and Applied Analysis
Lemma 1 (see [8]). Let 𝜃 be a lacunary sequence, and let
𝐿 𝑟 : 𝐶(𝑋) → 𝐶(𝑋) be a sequence of positive linear operators
satisfying
𝐿 𝑟 (𝑡V , 𝑥) 󴀀󴀤 𝑥] , (𝜃-equistat) ,
V = 0, 1, 2,
(6)
then for all 𝑓 ∈ 𝐶(𝑋),
𝐿 𝑟 (𝑓, 𝑥) 󴀀󴀤 𝑓, (𝜃-equistat) .
(7)
2. Lacunary Equistatistical Approximation
In this section, using the concept of Lacunary equistatistical
convergence, we give a Korovkin type result for a sequence
of positive linear operators defined on 𝐶(𝐼𝑚 ), the space of all
continuous real valued functions on the subset 𝐼𝑚 of R𝑚 , and
the real 𝑚-dimensional space. We first consider the case of
𝑚 = 2. Following [7] we can state the following theorem.
We turn to introducing some notation and the basic
definitions used in this paper. Throughout this paper 𝐼 =
[0, ∞). Let
Theorem 2. Let 𝜃 = {𝑘𝑟 } be a lacunary sequence, and let
{𝐿 𝑟 } be a sequence of positive linear operators from 𝐻𝑤2 into
𝐶𝐵 (𝐾). 𝐿 𝑟 is satisfying 𝐿 𝑟 (𝑓V ; 𝑥, 𝑦) 󴀀󴀤 𝑓V (𝑥, 𝑦) (𝜃-equistat),
V = 0, 1, 2, where 𝑓𝑘 (𝑢, V) ∈ 𝐻𝑤2 , 𝑘 = 0, 1, 2, 3,
𝐶 (𝐼) := {𝑓 : 𝑓 is a real-valued continuous function on 𝐼},
(8)
𝑓0 (𝑢, V) = 1,
and
𝐶𝐵 (𝐼) := {𝑓 ∈ 𝐶 (𝐼) : 𝑓 is bounded function on 𝐼} . (9)
Consider the space 𝐻𝑤 of all real-valued functions 𝑓 defined
on 𝐼 and satisfying
󵄨
󵄨󵄨
󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑦)󵄨󵄨󵄨 ≤ 𝑤 (𝑓;
󵄨󵄨 𝑥
𝑦 󵄨󵄨󵄨󵄨
󵄨󵄨
+
󵄨󵄨
󵄨󵄨) ,
󵄨󵄨 1 + 𝑥 1 + 𝑦 󵄨󵄨
(10)
𝑥,𝑦∈𝐼
|𝑥−𝑦|≤𝛿
for any 𝛿 > 0
𝑓 ∈ 𝐶𝐵 (𝐾) ,
(11)
(ii)
(iii)
≤
≤
𝑤2 (𝑓; 𝛿1󸀠 , 𝛿2 )
𝑤2 (𝑓; 𝛿1 , 𝛿2󸀠 )
(14)
V 2
𝑢 2
) +(
),
𝑢+1
V+1
(15)
Proof. Let (𝑥, 𝑦) ∈ 𝐾 be a fixed point, 𝑓 ∈ 𝐻𝑤2 , and assume
that (14) holds. For every 𝜀 > 0, there exist 𝛿1 , 𝛿2 > 0 such
that |𝑓(𝑢, V) − 𝑓(𝑥, 𝑦)| < 𝜀 holds for all (𝑢, V) ∈ 𝐾 satisfying
󵄨󵄨 𝑢
𝑥 󵄨󵄨󵄨
󵄨󵄨
󵄨󵄨 < 𝛿1 ,
−
󵄨󵄨
󵄨 𝑢 + 1 𝑥 + 1 󵄨󵄨
(12)
󵄨󵄨 V
𝑦 󵄨󵄨󵄨󵄨
󵄨󵄨
−
󵄨󵄨
󵄨 < 𝛿2 .
󵄨󵄨 V + 1 𝑦 + 1 󵄨󵄨󵄨
(16)
Let
(i) 𝑤2 (𝑓; 𝛿1 , ⋅) and 𝑤2 (𝑓; ⋅, 𝛿2 ) are nonnegative increasing
functions on [0, ∞),
+ 𝛿1󸀠󸀠 , 𝛿2 )
𝑤2 (𝑓; 𝛿1 , 𝛿2󸀠 + 𝛿2󸀠󸀠 )
V
,
V+1
then for all 𝑓 ∈ 𝐻𝑤2 ,
and also denote the valued of 𝐿𝑓 at a point (𝑥, 𝑦) ∈ 𝐾 is
denoted by 𝐿(𝑓; 𝑥, 𝑦) [10, 11].
𝑤2 (𝑓; 𝛿1 , 𝛿2 ) is the type of modulus of continuity for the
functions of two variables satisfying the following properties:
for any real numbers 𝛿1 , 𝛿2 , 𝛿1󸀠 , 𝛿2󸀠 , 𝛿1󸀠󸀠 , and 𝛿2󸀠󸀠 > 0,
𝑤2 (𝑓; 𝛿1󸀠
𝑓2 (𝑢, V) =
𝐿 𝑟 (𝑓; 𝑥, 𝑦) 󴀀󴀤 𝑓 (𝑥, 𝑦) , (𝜃-equistat) .
(see [9]). Let 𝐾 := 𝐼2 = [0, ∞) × [0, ∞), then the norm on
𝐶𝐵 (𝐾) is given by
󵄨
󵄨
󵄩󵄩 󵄩󵄩
󵄩󵄩𝑓󵄩󵄩 := sup 󵄨󵄨󵄨𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ,
(𝑥,𝑦)∈𝐾
𝑢
,
𝑢+1
𝑓3 (𝑢, V) = (
where 𝑤 is the modulus of continuity defined by
󵄨
󵄨
𝑤 (𝑓; 𝛿) := sup 󵄨󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑦)󵄨󵄨󵄨 ,
𝑓1 (𝑢, V) =
+
+
𝑤2 (𝑓; 𝛿1󸀠󸀠 , 𝛿2 ),
𝑤2 (𝑓; 𝛿1 , 𝛿2󸀠󸀠 ),
󵄨󵄨 𝑢
𝑥 󵄨󵄨󵄨
󵄨󵄨 < 𝛿1 ,
𝐾𝛿1 ,𝛿2 := {(𝑢, V) ∈ 𝐾 : 󵄨󵄨󵄨󵄨
−
󵄨 1 + 𝑢 1 + 𝑥 󵄨󵄨
Hence,
󵄨
󵄨󵄨
󵄨󵄨𝑓 (𝑢, V) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 =
󵄨
󵄨󵄨
󵄨󵄨𝑓 (𝑢, V) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨𝜒𝐾 (𝑢,V)
𝛿1 ,𝛿2
󵄨
󵄨
+ 󵄨󵄨󵄨𝑓 (𝑢, V) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨𝜒𝐾\𝐾
(iv) lim𝛿1 ,𝛿2 → 0 𝑤2 (𝑓; 𝛿1 , 𝛿2 ) = 0.
𝛿1 ,𝛿2
The space 𝐻𝑤2 is of all real-valued functions 𝑓 defined on
𝐾 and satisfying
󵄨
󵄨󵄨
󵄨󵄨𝑓 (𝑢, V) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨
󵄨󵄨 𝑢
𝑦 󵄨󵄨󵄨󵄨
𝑥 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 V
󵄨󵄨 , 󵄨󵄨
≤ 𝑤2 (𝑓; 󵄨󵄨󵄨󵄨
−
−
󵄨) .
󵄨 1 + 𝑢 1 + 𝑥 󵄨󵄨 󵄨󵄨 1 + V 1 + 𝑦 󵄨󵄨󵄨
(17)
󵄨󵄨 V
𝑦 󵄨󵄨󵄨󵄨
󵄨󵄨
−
󵄨󵄨
󵄨󵄨 < 𝛿2 } .
󵄨󵄨 V + 1 𝑦 + 1 󵄨󵄨
(13)
It is clear that any function in 𝐻𝑤2 is continuous and bounded
on 𝐾.
< 𝜀 + 2𝑀𝜒𝐾\𝐾
𝛿1 ,𝛿2
(𝑢,V)
(18)
(𝑢,V) ,
where 𝜒𝑃 denotes the characteristic function of the set 𝑃.
Observe that
𝜒𝐾\𝐾𝛿 ,𝛿 (𝑢, V) ≤
1 2
𝑦 2
𝑢
V
𝑥 2 1
1
(
+
(
−
)
−
).
𝛿12 1+ 𝑢 1+𝑥
𝛿22 V+1 𝑦+1
(19)
Abstract and Applied Analysis
3
Using (18), (19), and 𝑀 := ‖𝑓‖ we have
≤𝜀+
󵄨
󵄨󵄨
󵄨󵄨𝑓 (𝑢, V) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨
≤𝜀+
2
𝑦
2𝑀
𝑢
V
𝑥 2
{(
−
) +(
−
) },
2
𝛿
1+𝑢 1+𝑥
V+1 𝑦+1
(20)
where 𝛿 := min{𝛿1 , 𝛿2 }.
By the linearity and positivity of the operators {𝐿 𝑟 } and by
(18), we have
𝐿 𝑟 ((𝑓1 −
+
4𝑀 󵄨󵄨
󵄨
󵄨𝐿 (𝑓 ; 𝑥, 𝑦) − 𝑓2 (𝑥, 𝑦)󵄨󵄨󵄨
𝛿2 󵄨 𝑟 2
+
4𝑀 󵄨󵄨
󵄨
󵄨𝐿 (𝑓 ; 𝑥, 𝑦) − 𝑓1 (𝑥, 𝑦)󵄨󵄨󵄨
𝛿2 󵄨 𝑟 1
+ (𝜀 + 𝑀 +
=
2
2
𝑢
V
𝑓0 ) + (𝑓2 −
𝑓0 ) ; 𝑥, 𝑦)
1+𝑢
1+V
≤ 𝐿 𝑟 (𝑓3 ; 𝑥, 𝑦)
4𝑀 󵄨󵄨
󵄨
󵄨𝐿 (𝑓 ; 𝑥, 𝑦) − 𝑓2 (𝑥, 𝑦)󵄨󵄨󵄨
𝛿2 󵄨 𝑟 2
4𝑀 󵄨󵄨
󵄨
󵄨𝐿 (𝑓 ; 𝑥, 𝑦) − 𝑓1 (𝑥, 𝑦)󵄨󵄨󵄨
𝛿2 󵄨 𝑟 1
󵄨
󵄨
+ 𝜀 + 𝑁 󵄨󵄨󵄨𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓0 (𝑥, 𝑦)󵄨󵄨󵄨 ,
+
𝑦
𝑥
− 2[
𝐿 𝑟 (𝑓1 ; 𝑥, 𝑦) +
𝐿 (𝑓 ; 𝑥, 𝑦)] 𝐿 𝑟 (𝑓3 ; 𝑥, 𝑦)
1+𝑥
1+𝑦 𝑟 2
𝑦
𝑥
𝐿 𝑟 (𝑓1 ; 𝑥, 𝑦) +
𝐿 (𝑓 ; 𝑥, 𝑦)]
1+𝑥
1+𝑦 𝑟 2
+ [(
𝑦 2
𝑥 2
) +(
) ] 𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) .
1+𝑥
1+𝑦
(22)
where 𝑁 := 𝜀 + 𝑀 + 4𝑀/𝛿2 . For a given 𝜇 > 0, choose 𝜀 > 0
such that 𝜀 < 𝜇. Define the following sets:
(21)
Hence, we get
󵄨󵄨
󵄨
󵄨󵄨𝐿 𝑟 (𝑓; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨
󵄨
󵄨
≤ 𝐿 𝑟 (󵄨󵄨󵄨𝑓 (𝑢, V) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ; 𝑥, 𝑦)
󵄨󵄨
󵄨
󵄨
+ 󵄨󵄨󵄨𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 󵄨󵄨󵄨𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓0 (𝑥, 𝑦)󵄨󵄨󵄨
4𝑀 󵄨󵄨
󵄨
) 󵄨𝐿 (𝑓 ; 𝑥, 𝑦) − 𝑓0 (𝑥, 𝑦)󵄨󵄨󵄨
𝛿2 󵄨 𝑟 0
2𝑀 󵄨󵄨
󵄨
󵄨𝐿 (𝑓 ; 𝑥, 𝑦) − 𝑓3 (𝑥, 𝑦)󵄨󵄨󵄨
𝛿2 󵄨 𝑟 3
+
− 2[
2𝑀 󵄨󵄨
󵄨
󵄨𝐿 (𝑓 ; 𝑥, 𝑦) − 𝑓3 (𝑥, 𝑦)󵄨󵄨󵄨
𝛿2 󵄨 𝑟 3
󵄨
󵄨
𝐷𝜇 (𝑥, 𝑦) := {𝑚 ∈ N : 󵄨󵄨󵄨𝐿 𝑚 (𝑓; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ≥ 𝜇} ,
󵄨
󵄨 𝜇−𝜀
𝐷𝜇V (𝑥, 𝑦) := {𝑚 ∈ N : 󵄨󵄨󵄨𝐿 𝑚 (𝑓; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ≥
},
4𝑁
(23)
where V = 0, 1, 2, 3. Then from (22) we clearly have
3
𝐷𝜇 (𝑥, 𝑦) ⊆ ⋃𝐷𝜇V (𝑥, 𝑦) .
(24)
V=0
2
2𝑀
𝑢
𝑥
≤ 𝐿 𝑟 (𝜀 + 2 { (
−
)
𝛿
1+𝑢 1+𝑥
Therefore define the following real valued functions:
𝑦 2
V
+(
−
) } ; 𝑥, 𝑦)
V+1 1+𝑦
󵄨
󵄨
+ 𝑀 󵄨󵄨󵄨𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓0 (𝑥, 𝑦)󵄨󵄨󵄨
≤ 𝐿 𝑟 (𝜀 +
2
𝑦
⋅ 𝑓 ) } ; 𝑥, 𝑦)
1+𝑦 0
󵄨
󵄨
+ 𝑀 󵄨󵄨󵄨𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓0 (𝑥, 𝑦)󵄨󵄨󵄨
2
𝑦
𝑓 ) ; 𝑥, 𝑦)
1+𝑦 0
󵄨
󵄨
+ 𝑀 󵄨󵄨󵄨𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓0 (𝑥, 𝑦)󵄨󵄨󵄨
1
ℎ𝑟
󵄨󵄨
󵄨
󵄨󵄨{𝑚 ∈ 𝐼 : 󵄨󵄨󵄨𝐿 (𝑓 ; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ≥ 𝜇 − 𝜀 }󵄨󵄨󵄨 ,
󵄨󵄨
󵄨
𝑟 󵄨 𝑚
V
󵄨
4𝑁 󵄨󵄨
󵄨
(25)
where V = 0, 1, 2, 3. Then by the monotonicity and (24) we get
3
V
𝑠𝑟,𝜇 (𝑥, 𝑦) ≤ ∑ 𝑠𝑟,𝜇
(𝑥, 𝑦)
(26)
𝑣=0
2
2𝑀
𝑥
= 𝐿 𝑟 (𝜀; 𝑥, 𝑦) + 𝐿 𝑟 ( 2 (𝑓1 −
𝑓0 )
𝛿
1+𝑥
+ (𝑓2 −
1 󵄨󵄨
󵄨
󵄨
󵄨
󵄨{𝑚 ∈ 𝐼𝑟 : 󵄨󵄨󵄨𝐿 𝑚 (𝑓; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ≥ 𝜇}󵄨󵄨󵄨 ,
ℎ𝑟 󵄨
V
𝑠𝑟,𝜇
(𝑥, 𝑦)
:=
2
2𝑀
𝑥
{(𝑓1 −
⋅ 𝑓0 )
2
𝛿
1+𝑥
+ (𝑓2 −
𝑠𝑟,𝜇 (𝑥, 𝑦) :=
for all 𝑥 ∈ 𝑋, and this implies the inequality
3
󵄩󵄩
󵄩
󵄩󵄩
󵄩V
󵄩󵄩𝑠𝑟,𝜇 (⋅)󵄩󵄩󵄩 ≤ ∑ 󵄩󵄩󵄩𝑠𝑟,𝜇
󵄩
󵄩
󵄩𝐾
󵄩 (⋅)󵄩󵄩𝐾 .
(27)
V=0
Taking limit in (27) as 𝑟 → ∞ and using (14) we have
󵄩󵄩
lim 󵄩󵄩𝑠𝑟,𝜇
𝑟 → ∞󵄩
󵄩
(⋅)󵄩󵄩󵄩󵄩𝐾 = 0.
(28)
4
Abstract and Applied Analysis
Then for all 𝑓 ∈ 𝐻𝑤2 , we conclude that
𝐿 𝑟 (𝑓; 𝑥, 𝑦) 󴀀󴀤 𝑓 (𝑥, 𝑦) , (𝜃-equistat) .
Lemma 4. Let 𝜃 = {𝑘𝑟 } be a lacunary sequence, and let
(29)
𝐵𝑛 (𝑓; 𝑥, 𝑦) =
Now replace 𝐼2 by 𝐼𝑚 = [0, ∞) × ⋅ ⋅ ⋅ × [0, ∞) and by
an induction, we consider the modulus of continuity type
function 𝑤𝑚 as in the function 𝑤2 . Then let 𝐻𝑤𝑚 be the space
of all real-valued functions 𝑓 satisfying
󵄨
󵄨󵄨
󵄨󵄨𝑓 (𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ) − 𝑓 (𝑥1 , 𝑥2 , . . . , 𝑥𝑚 )󵄨󵄨󵄨
󵄨󵄨󵄨 𝑢
󵄨󵄨󵄨 𝑢
𝑥𝑚 󵄨󵄨󵄨󵄨
𝑥1 󵄨󵄨󵄨󵄨
≤ 𝑤2 (𝑓 ; 󵄨󵄨󵄨 1 −
󵄨󵄨 , . . . , 󵄨󵄨󵄨 𝑚 −
󵄨󵄨) .
󵄨󵄨 𝑢𝑚 + 1 𝑥𝑚 + 1 󵄨󵄨
󵄨󵄨 𝑢1 + 1 𝑥1 + 1 󵄨󵄨
(30)
Therefore, using a similar technique in the proof of
Lemma 1 one can obtain the following result immediately.
1
𝑛
(1 + 𝑥) (1 + 𝑦)
𝑛
𝑛
𝑘=0 𝑙=0
V = 0, 1, 2, . . . , 𝑚 + 1,
𝐵𝑛 (𝑓V ; 𝑥, 𝑦) 󴀀󴀤 𝑓V (𝑥, 𝑦) , (𝜃-equistat) ,
𝑓1 (𝑢, V) =
𝑢
,
𝑢+1
𝑓2 (𝑢, V) =
]
,
]+1
𝑓3 (𝑢, V) = (
𝑓0 (𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ) = 1,
𝑢1
𝑓1 (𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ) =
,
𝑢1 + 1
..
.
𝑓𝑚 (𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ) =
𝑢𝑚
,
𝑢𝑚 + 1
2
𝑢
𝑓𝑚+1 (𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ) = ( 1 )
𝑢1 + 1
2
2
𝑢
𝑢
+ ( 2 ) + ⋅⋅⋅ + ( 𝑚 ) .
𝑢2 + 1
𝑢𝑚 + 1
(32)
𝐿 𝑟 (𝑓; 𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ) 󴀀󴀤 𝑓 (𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ) , (𝜃-equistat) .
(33)
Assume that 𝐼 = [0, ∞), 𝐾 := 𝐼×𝐼. One considers the following
positive linear operators defined on 𝐻𝑤2 (𝐾):
1
𝑛
(1 + 𝑥) (1 + 𝑦)
𝑛
𝑛
𝑢 2
] 2
) +(
),
𝑢+1
]+1
(36)
then for all 𝑓 ∈ 𝐻𝑤2 ,
𝐵𝑛 (𝑓; 𝑥, 𝑦) 󴀀󴀤 𝑓 (𝑥, 𝑦) , (𝜃-equistat) .
(37)
Proof. Assume that (36) holds, and let 𝑓 ∈ 𝐻𝑤2 . Since
𝑛
𝑛
(1 + 𝑥)𝑛 = ∑ ( ) 𝑥𝑘 ,
𝑘
𝑘=0
𝑛
𝑛
𝑛
(1 + 𝑦) = ∑ ( ) 𝑦𝑙 ,
𝑙
(38)
𝑙=0
it is clear that, for all 𝑛 ∈ N,
Then for all 𝑓 ∈ 𝐻𝑤𝑚 ,
𝐵𝑛 (𝑓; 𝑥, 𝑦) =
V = 0, 1, 2, 3,
𝑓0 (𝑢, V) = 1,
(31)
where 𝑓𝑘 (𝑢1 , 𝑢2 , . . . , 𝑢𝑚 ) ∈ 𝐻𝑤𝑚 , 𝑘 = 0, 1, 2, . . . 𝑚 + 1,
𝑘
𝑙
𝑛 𝑛
,
) ( ) ( ) 𝑥 𝑘 𝑦𝑙
𝑘
𝑙
𝑛 − 𝑘+1 𝑛 − 𝑙+1
(35)
be a sequence of positive linear operators from 𝐻𝑤2 into 𝐶𝐵 (𝐾).
If 𝐵𝑛 is satisfying
Theorem 3. Let 𝜃 = {𝑘𝑟 } be a lacunary sequence, and let
{𝐿 𝑟 } be a sequence of positive linear operators from 𝐻𝑤𝑚 into
𝐶𝐵 (𝐼𝑚 ). 𝐿 𝑟 is satisfying
𝐿 𝑟 (𝑓V ; 𝑥, 𝑦) 󴀀󴀤 𝑓V (𝑥, 𝑦) , (𝜃-equistat) ,
𝑛
× ∑ ∑𝑓 (
𝑛
× ∑ ∑𝑓 (
𝑘=0 𝑙=0
𝑘
𝑙
𝑛 𝑛
,
) ( ) ( ) 𝑥 𝑘 𝑦𝑙 ,
𝑘
𝑙
𝑛 − 𝑘+1 𝑛 − 𝑙+1
(34)
where 𝑓 ∈ 𝐻𝑤2 , (𝑥, 𝑦) ∈ 𝐾 and 𝑛 ∈ N.
𝐵𝑛 (𝑓0 ; 𝑥, 𝑦) =
=
𝑛 𝑛
1
𝑛 𝑘 𝑛 𝑘
𝑛 ∑ ∑ (𝑘) 𝑥 ( 𝑙 ) 𝑦
(1 + 𝑥) (1 + 𝑦) 𝑘=0 𝑙=0
𝑛
𝑛
𝑛
1
𝑛 𝑘
𝑛
(
(
)
( ) 𝑦𝑘 = 1.
)
𝑥
∑
∑
𝑛
𝑙
(1+𝑥)𝑛 (1+𝑦) 𝑘=0 𝑘
𝑙=0
(39)
Now, by assumption we have
𝐵𝑛 (𝑓0 ; 𝑥, 𝑦) 󴀀󴀤 𝑓0 (𝑥, 𝑦) , (𝜃-equistat) .
(40)
Abstract and Applied Analysis
5
Using the definition of 𝐵𝑛 , we get
To see this, by the definition of 𝐵𝑛 , we first write
𝐵𝑛 (𝑓1 ; 𝑥, 𝑦)
=
𝐵𝑛 (𝑓3 ; 𝑥, 𝑦) =
1
𝑛
(1 + 𝑥) (1 + 𝑦)
𝑛
𝑛 𝑛
1
𝑛 𝑛
∑
∑ ( ) ( ) 𝑥 𝑘 𝑦𝑙
𝑛
𝑛
𝑙
(1 + 𝑥) (1 + 𝑦) 𝑘=0 𝑙=0 𝑘
×[
𝑛
𝑘⟋ (𝑛 − 𝑘 + 1)
𝑛
𝑛
( ) 𝑥𝑘 ∑ ( ) 𝑦𝑙
𝑙
𝑘
−
𝑘
+
1))
+
1
(𝑘⟋
(𝑛
𝑘=1
𝑙=0
𝑛
×∑
=
𝑛
𝑛
𝑘
1
𝑛
𝑛 𝑘
( ) 𝑦𝑙
(
)
𝑥
∑
∑
𝑛
𝑛
𝑙
𝑘
(1 + 𝑥) (1 + 𝑦) 𝑘=1 𝑛 + 1
𝑙=0
=
1 𝑛−1 𝑘 + 1
𝑛
(
) 𝑥𝑘+1
𝑛∑
𝑘
+
1
(1 + 𝑥) 𝑘=0 𝑛 + 1
=
𝑥 𝑛−1 𝑘 + 1
𝑛
(
) 𝑥𝑘 .
𝑛∑
𝑘
+
1
(1 + 𝑥) 𝑘=0 𝑛 + 1
=
+
Since
𝑛
𝑛
𝑛−1
(
)=(
)
𝑘+1
𝑘
𝑘+1
=
=
=
∞
𝑛
𝑙2
1
𝑛 𝑘
𝑛
(
( ) 𝑦𝑙
)
𝑥
∑
∑
𝑛
𝑛
2
𝑘
𝑙
(1+𝑥) (1+𝑦) 𝑘=0
𝑙=1 (𝑛+1)
𝑛
𝑘
1
𝑥𝑘
∑
𝑛
(1 + 𝑥) 𝑘=1 (𝑛 + 1)2
+
𝑛
𝑘 (𝑘 − 1) 𝑛 𝑘
1
( )𝑥
∑
𝑛
(1 + 𝑥) 𝑘=2 (𝑛 + 1)2 𝑘
+
𝑛
𝑙 (𝑙 − 1) 𝑛 𝑙
1
( )𝑦
∑
𝑛
(1 + 𝑦) 𝑙=2 (𝑛 + 1)2 𝑙
+
𝑛
𝑙
1
𝑦𝑙 .
𝑛∑
(1 + 𝑦) 𝑙=1 (𝑛 + 1)2
(42)
we get
𝐵𝑛 (𝑓1 ; 𝑥, 𝑦) =
𝑛
𝑛
𝑘2
1
𝑛 𝑘
𝑛
(
( ) 𝑦𝑙
)
𝑥
∑
∑
𝑛
𝑛
2 𝑘
𝑙
(1 + 𝑥) (1 + 𝑦) 𝑘=1 (𝑛 + 1)
𝑙=0
(41)
=
𝑘2
𝑙2
+
]
(𝑛 + 1)2 (𝑛 + 1)2
(48)
𝑥 𝑛−1 𝑘 + 1 𝑛
𝑛−1 𝑘
(
)𝑥
∑
𝑘
(1 + 𝑥)𝑛 𝑘=0 𝑛 + 1 𝑘 + 1
Then,
𝐵𝑛 (𝑓3 ; 𝑥, 𝑦) =
𝑥 𝑛−1 𝑛
𝑛−1 𝑘
(
)𝑥
∑
𝑘
(1 + 𝑥)𝑛 𝑘=0 𝑛 + 1
𝑛 (𝑛 − 1) 𝑥2
(𝑛 + 1)2 (𝑥 + 1)2
+
𝑛−1
𝑥
𝑛
𝑛−1 𝑘
⋅
⋅∑(
)𝑥
𝑘
(1 + 𝑥) (1 + 𝑥)𝑛−1 𝑛 + 1 𝑘=0
𝑛
𝑥
(
).
𝑛+1 𝑥+1
(43)
+
󵄨󵄨
𝑥 󵄨󵄨󵄨 𝑛
󵄨󵄨 .
󵄨󵄨
−
1
󵄨󵄨
𝑥 + 1 󵄨󵄨 𝑛 + 1
󵄨
(50)
Since
lim
𝑛→∞
𝑛 (𝑛 − 1)
= 1,
(𝑛 + 1)2
lim
𝑛 → ∞ (𝑛
𝑛
= 0,
+ 1)2
(51)
we get
(46)
󵄩
󵄩󵄩 1
󵄨󵄩󵄩
󵄩 󵄨
󵄨
󵄨
lim 󵄩󵄩󵄩 󵄨󵄨󵄨{𝑚 ∈ 𝐼𝑟 : 󵄨󵄨󵄨𝐵𝑛 (𝑓3 ; 𝑥, 𝑦) − 𝑓3 (𝑥, 𝑦)󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨󵄩󵄩󵄩 = 0.
𝑟→∞ 󵄩ℎ
󵄩󵄩
󵄩 𝑟
(52)
(47)
Thus 𝐵𝑛 (𝑓3 ; 𝑥, 𝑦) 󴀀󴀤 𝑓3 (𝑥, 𝑦), (𝜃-equistat). Therefore we
obtain that for all 𝑓 ∈ 𝐻𝑤2 , 𝐵𝑛 (𝑓; 𝑥, 𝑦) 󴀀󴀤 𝑓(𝑥, 𝑦),
(𝜃-equistat).
Hence we have
Also we have
𝐵𝑛 (𝑓3 ; 𝑥, 𝑦) 󴀀󴀤 𝑓3 (𝑥, 𝑦) , (𝜃-equistat) .
𝑛 (𝑛 − 1)
2
(𝑦 + 1) (𝑛 + 1)
2
󵄨󵄨
󵄨󵄨
𝑦2
𝑥2
󵄨󵄨 𝑛 (𝑛 − 1)
󵄨󵄨
󵄨
󵄨󵄨
+
)
−
1
󵄨
2
2
2
󵄨
󵄨󵄨
(𝑥 + 1)
(𝑦 + 1) 󵄨󵄨 (𝑛 + 1)
󵄨
󵄨
󵄨
𝑦 󵄨󵄨󵄨
𝑛
󵄨󵄨󵄨 𝑥
+
+
󵄨.
2 󵄨󵄨󵄨 𝑥 + 1
𝑦
+ 1 󵄨󵄨󵄨
(𝑛 + 1) 󵄨
(44)
󵄩
󵄩󵄩 1
󵄨󵄩󵄩
󵄩 󵄨
󵄨
󵄨
lim 󵄩󵄩󵄩 󵄨󵄨󵄨{𝑚 ∈ 𝐼𝑟 : 󵄨󵄨󵄨𝐵𝑛 (𝑓1 ; 𝑥, 𝑦) − 𝑓1 (𝑥, 𝑦)󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨󵄩󵄩󵄩 = 0.
𝑟→∞ 󵄩ℎ
󵄩󵄩
󵄩 𝑟
(45)
𝐵𝑛 (𝑓2 ; 𝑥, 𝑦) 󴀀󴀤 𝑓2 (𝑥, 𝑦) , (𝜃-equistat) .
𝑦2
=(
The fact that lim𝑛 → ∞ (𝑛/(𝑛 + 1)) = 1 and using a similar
technique as in the proof of Lemma 1, we get
(49)
which implies that
󵄨
󵄨󵄨
󵄨󵄨𝐵𝑛 (𝑓3 ; 𝑥, 𝑦) − 𝑓3 (𝑥, 𝑦)󵄨󵄨󵄨
So, we have
󵄨
󵄨󵄨
󵄨󵄨𝐵𝑛 (𝑓1 ; 𝑥, 𝑦) − 𝑓1 (𝑥, 𝑦)󵄨󵄨󵄨 =
𝑦
𝑥
𝑛
𝑛
+
(𝑛 + 1)2 (𝑥 + 1) (𝑛 + 1)2 𝑦 + 1
6
Abstract and Applied Analysis
3. Rates of Lacunary
Equistatistical Convergence
and hence
In this section we study the order of lacunary equi-statistical
convergence of a sequence of positive linear operators acting
on 𝐻𝑤2 (𝐾), where 𝐾 = 𝐼𝑚 . To achieve this we first consider
the case of 𝑚 = 2.
Definition 5. A sequence {𝑓𝑟 } is called lacunary equistatistically convergent to a function 𝑓 with rate 0 < 𝛽 < 1 if
for every 𝜀 > 0,
lim
𝑟→∞
𝑠𝑟,𝜀 (𝑥, 𝑦)
= 0,
𝑟−𝛽
(53)
where 𝑠𝑟,𝜀 (𝑥, 𝑦) is given in Lemma 1. In this case it is denoted
by
𝑓𝑟 − 𝑓 = 𝑜 (𝑟−𝛽 ) , (𝜃-equistat)
on 𝐾 = 𝐼 × 𝐼.
(54)
󵄩
󵄩󵄩
󵄩󵄩𝑠𝑟,𝜀 (𝑥, 𝑦)󵄩󵄩󵄩𝐻𝑤 (𝐾)
2
𝑟−𝛽
𝑔𝑟 − 𝑔 = 𝑜 (𝑟−𝛽2 ) , (𝜃-equistat) .
Now we give the rate of lacunary equi-statistical convergence of a positive linear operators 𝐿 𝑟 (𝑓; 𝑥, 𝑦) to 𝑓(𝑥, 𝑦) with
the help of modulus of continuity.
Theorem 7. Let 𝐾 = 𝐼 × 𝐼, and let 𝐿 𝑟 : 𝐻𝑤2 (𝐾) → 𝐻𝑤2 (𝐾)
be a sequence of positive linear operators. Assume that
(i) 𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓0 = 𝑜(𝑟−𝛽1 ), (𝜃-equistat) on 𝐾,
(ii) 𝑤(𝑓; 𝛿𝑟,𝑥 , 𝛿𝑟,𝑦 ) = 𝑜(𝑟−𝛽2 ), (𝜃-equistat) on 𝐾 with
(𝑓𝑟 + 𝑔𝑟 ) − (𝑓 + 𝑔) = 𝑜 (𝑟 ) , (𝜃-equistat) ,
(56)
where 𝛽 = min{𝛽1 , 𝛽2 }.
𝑦 2
V
= √ 𝐿 𝑟 ((
−
) , 𝑦).
1+V 1+𝑦
(60)
Proof. Assume that 𝑓𝑟 − 𝑓 = 𝑜(𝑟 ), (𝜃-equistat) and 𝑔𝑟 −
𝑔 = 𝑜(𝑟−𝛽2 ), (𝜃-equistat) on 𝐾. For all 𝜀 > 0, consider the
following functions:
𝑠𝑟,𝜀 (𝑥, 𝑦)
1 󵄨󵄨
󵄨
󵄨
󵄨
󵄨{𝑛 ∈ 𝐼𝑟 : 󵄨󵄨󵄨(𝑓𝑛 + 𝑔𝑛 ) (𝑥, 𝑦) − (𝑓 + 𝑔) (𝑥, 𝑦)󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨 ,
ℎ𝑟 󵄨
󵄨󵄨
󵄨
󵄨󵄨{𝑛 ∈ 𝐼 : 󵄨󵄨󵄨𝑓 (𝑥) − 𝑓 (𝑥)󵄨󵄨󵄨 ≥ 𝜀 }󵄨󵄨󵄨 ,
󵄨󵄨
󵄨
𝑟 󵄨 𝑛
󵄨
2 󵄨󵄨
󵄨
𝜀 󵄨󵄨
1 󵄨󵄨󵄨
2
󵄨󵄨{𝑛 ∈ 𝐼𝑟 : 󵄨󵄨󵄨󵄨𝑔𝑛 (𝑥) − 𝑔 (𝑥)󵄨󵄨󵄨󵄨 ≥ }󵄨󵄨󵄨 .
𝑠𝑟,𝜀
(𝑥, 𝑦) :=
󵄨
ℎ𝑟 󵄨
2 󵄨󵄨
1
ℎ𝑟
Then we have
󵄩󵄩
󵄩
󵄩󵄩 1
󵄩
󵄩󵄩𝑠𝑟,𝜀 (𝑥, 𝑦)󵄩󵄩󵄩𝐻𝑤 (𝐾)
󵄩󵄩𝑠𝑟,𝜀 (𝑥, 𝑦)󵄩󵄩󵄩
2
󵄩
󵄩
lim
= lim
𝑟→∞
𝑟→∞
𝑟−𝛽
𝑟−𝛽
󵄩󵄩 2
󵄩󵄩
󵄩󵄩󵄩𝑠𝑟,𝜀 (𝑥, 𝑦)󵄩󵄩󵄩
+ lim
,
𝑟→∞
𝑟−𝛽
𝑠𝑟,𝜀 (𝑥, 𝑦)
≤
𝑟−𝛽
≤
𝐿 𝑟 (𝑓; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦) = 𝑜 (𝑟−𝛽 ) , (𝜃-equistat) 𝑜𝑛 𝐾, (61)
where 𝛽 = min{𝛽1 , 𝛽2 }.
−𝛽1
1
𝑠𝑟,𝜀
(𝑥, 𝑦) :=
𝛿𝑟,𝑦
𝑢
𝑥 2
−
) , 𝑥),
1+𝑢 1+𝑥
Then
−𝛽
:=
𝛿𝑟,𝑥 = √ 𝐿 𝑟 ((
(55)
Then one has
󵄩
󵄩󵄩 2
󵄩󵄩𝑠𝑟,𝜀 (𝑥, 𝑦)󵄩󵄩󵄩
󵄩𝐻𝑤2 (𝐾)
󵄩
.
−𝛽
2
𝑟
(59)
Taking limit as 𝑟 → ∞ and using the assumption complete
the proof.
Lemma 6. Let {𝑓𝑟 } and {𝑔𝑟 } be two sequences of functions in
𝐻𝑤2 (𝐾), with
𝑓𝑟 − 𝑓 = 𝑜 (𝑟−𝛽1 ) , (𝜃-equistat) ,
󵄩
󵄩󵄩 1
󵄩󵄩𝑠𝑟,𝜀 (𝑥, 𝑦)󵄩󵄩󵄩
󵄩𝐻𝑤2 (𝐾)
󵄩
≤
+
−𝛽
1
𝑟
1
(𝑥, 𝑦)
𝑠𝑟,𝜀
𝑟−𝛽
1
𝑠𝑟,𝜀
(𝑥, 𝑦)
𝑟−𝛽1
+
+
2
(𝑥, 𝑦)
𝑠𝑟,𝜀
(57)
Proof. Let 𝑓 ∈ 𝐻𝑤2 (𝐾) and 𝑥 ∈ 𝐾. Use
󵄨󵄨
󵄨
󵄨󵄨𝐿 𝑟 (𝑓; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨
󵄨
󵄨
≤ 𝐿 𝑟 (󵄨󵄨󵄨𝑓 (𝑢, V) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ; 𝑥, 𝑦)
󵄨󵄨
󵄨
󵄨
+ 󵄨󵄨󵄨𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 󵄨󵄨󵄨𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨
󵄨󵄨 𝑢
𝑦 󵄨󵄨󵄨󵄨
𝑥 󵄨󵄨󵄨 󵄨󵄨󵄨󵄨 V
󵄨󵄨 , 󵄨󵄨
≤ 𝐿 𝑟 (𝑤2 (𝑓; 󵄨󵄨󵄨󵄨
−
−
󵄨) ; 𝑥, 𝑦)
󵄨 1 + 𝑢 1 + 𝑥 󵄨󵄨 󵄨󵄨 1 + V 1 + 𝑦 󵄨󵄨󵄨
󵄨󵄨
󵄨
󵄨
+ 󵄨󵄨󵄨𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 󵄨󵄨󵄨𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓0 (𝑥, 𝑦)󵄨󵄨󵄨
≤ (1 + 𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦)) 𝑤2 (𝑓; 𝛿𝑟,𝑥 , 𝛿𝑟,𝑦 )
󵄨
󵄨
+ 𝑀 󵄨󵄨󵄨𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓0 (𝑥, 𝑦)󵄨󵄨󵄨
󵄨
󵄨
= 2𝑤2 (𝑓; 𝛿𝑟,𝑥 , 𝛿𝑟,𝑦 ) + 𝑀 󵄨󵄨󵄨𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓0 (𝑥, 𝑦)󵄨󵄨󵄨
󵄨
󵄨
+ 𝑤2 (𝑓; 𝛿𝑟,𝑥 , 𝛿𝑟,𝑦 ) 󵄨󵄨󵄨𝐿 𝑟 (𝑓0 ; 𝑥, 𝑦) − 𝑓0 (𝑥, 𝑦)󵄨󵄨󵄨 ,
(58)
where 𝑀 = ‖𝑓‖𝐻𝑤 (𝐾) . Using inequality (62), conditions (i)
2
and (ii) we get
𝑟−𝛽
2
𝑠𝑟,𝜀
(𝑥, 𝑦)
𝑟−𝛽2
lim
,
(62)
󵄩
1 󵄩󵄩󵄩󵄩 1 󵄨󵄨
󵄨󵄩󵄩
󵄨
󵄨
󵄩󵄩 󵄨󵄨{𝑟 ∈ 𝐼𝑟 : 󵄨󵄨󵄨𝐿 𝑟 (𝑓; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 ≥ 𝜀}󵄨󵄨󵄨󵄩󵄩󵄩 = 0,
󵄩󵄩
󵄩󵄩 ℎ𝑟
(63)
𝑟 → ∞ 𝑟−𝛽
Abstract and Applied Analysis
7
References
so we have
−𝛽
𝐿 𝑟 (𝑓; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦) = 𝑜 (𝑟 ) , (𝜃-equistat) on 𝐾.
(64)
Finally we give the rate of lacunary equi-statistical convergence for the operators 𝐿 𝑟 (𝑓, 𝑥) by using the Peetre’s 𝐾functional in the space 𝐻𝑤2 (𝐾). The Peetre 𝐾-functional of
function 𝑓 ∈ 𝐻𝑤2 (𝐾) is defined by
󵄩
󵄩
󵄩 󵄩
𝐾 (𝑓; 𝛿𝑟,𝑥 , 𝛿𝑟,𝑦 ) = inf {󵄩󵄩󵄩𝑓 − 𝑔󵄩󵄩󵄩𝐶𝐵 (𝐾) + 𝛿󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐶𝐵 (𝐾) } ,
𝑔∈𝐻𝑤 (𝐾)
2
(65)
where
󵄩󵄩 󵄩󵄩
󵄨
󵄨
󵄩󵄩𝑓󵄩󵄩𝐶𝐵 (𝐾) = sup 󵄨󵄨󵄨𝑓 (𝑥, 𝑦)󵄨󵄨󵄨 .
𝑥,𝑦
∈𝐾
( )
(66)
Theorem 8. Let 𝑓 ∈ 𝐻𝑤2 (𝐾) and {𝐾(𝑓; 𝛿𝑟,𝑥 , 𝛿𝑟,𝑦 )} be the
sequence of Peetre’s K-functional. If
󵄩󵄩
󵄩󵄩
𝑥
𝛿𝑟,𝑥 = 󵄩󵄩󵄩󵄩𝐿 𝑟 ((𝑓1 −
) ; 𝑥, 𝑦)󵄩󵄩󵄩󵄩
1+𝑥
󵄩
󵄩𝐶𝐵 (𝐾)
󵄩󵄩
󵄩󵄩
2
𝑥
󵄩
󵄩
󵄩󵄩 󵄩󵄩
+ 󵄩󵄩󵄩𝐿 𝑟 ((𝑓1 −
,
) ; 𝑥, 𝑦)󵄩󵄩󵄩
󵄩𝑔󵄩
󵄩󵄩
󵄩󵄩𝐶𝐵 (𝐾) 󵄩 󵄩𝐶𝐵 (𝐾)
1+𝑥
󵄩󵄩󵄩
󵄩󵄩󵄩
𝑦
𝛿𝑟,𝑦 = 󵄩󵄩󵄩𝐿 𝑟 ((𝑓2 −
) ; 𝑥, 𝑦)󵄩󵄩󵄩
1+𝑦
󵄩󵄩
󵄩󵄩𝐶𝐵 (𝐾)
(67)
󵄩󵄩
󵄩󵄩
2
𝑦
󵄩
󵄩
󵄩󵄩 󵄩󵄩
+ 󵄩󵄩󵄩󵄩𝐿 𝑟 ((𝑓2 −
,
) ; 𝑥, 𝑦)󵄩󵄩󵄩󵄩
󵄩𝑔󵄩
1+𝑦
󵄩󵄩
󵄩󵄩𝐶𝐵 (𝐾) 󵄩 󵄩𝐶𝐵 (𝐾)
󵄩󵄩
󵄩󵄩
= 0, (𝜃-equistat)
󵄩󵄩
󵄩󵄩
= 0, (𝜃-equistat)
lim 󵄩𝛿 󵄩
𝑟 → ∞ 󵄩 𝑟,𝑥 󵄩
lim 󵄩󵄩𝛿𝑟,𝑦 󵄩󵄩󵄩
𝑟→∞ 󵄩
on 𝑥, 𝑦 ∈ 𝐾, then
󵄩󵄩
󵄩
󵄩󵄩𝐿 𝑟 (𝑓; 𝑥, 𝑦) − 𝑓 (𝑥, 𝑦)󵄩󵄩󵄩𝐶𝐵 (𝐾) ≤ 𝐾 (𝑓; 𝛿𝑟,𝑥 , 𝛿𝑟,𝑦 ) .
Proof. For each 𝑔 ∈ 𝐻𝑤2 (𝐾), we get
󵄩󵄩
󵄩
󵄩󵄩𝐿 𝑟 (𝑔; 𝑥, 𝑦) − 𝑔 (𝑥, 𝑦)󵄩󵄩󵄩𝐶𝐵 (𝐾)
(68)
󵄩󵄩
󵄩󵄩
𝑥
󵄩 󵄩
≤ 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐶𝐵 (𝐾) (󵄩󵄩󵄩󵄩𝐿 𝑟 ((𝑓1 −
) ; 𝑥, 𝑦)󵄩󵄩󵄩󵄩
1+𝑥
󵄩
󵄩𝐶𝐵 (𝐾)
󵄩󵄩
󵄩󵄩
𝑥 2
󵄩
󵄩
+ 󵄩󵄩󵄩𝐿 𝑟 ((𝑓1 −
) ; 𝑥, 𝑦)󵄩󵄩󵄩 )
󵄩󵄩
󵄩󵄩
1+𝑥
󵄩󵄩
󵄩󵄩
𝑦
󵄩 󵄩
󵄩
󵄩
+ 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐶𝐵 (𝐾) (󵄩󵄩󵄩𝐿 𝑟 (𝑓2 −
)
) ; 𝑥, 𝑦󵄩󵄩󵄩
󵄩󵄩
󵄩󵄩𝐶 (𝐾)
1+𝑦
𝐵
󵄩󵄩
󵄩󵄩
𝑦 2
󵄩
󵄩
+ 󵄩󵄩󵄩󵄩𝐿 𝑟 ((𝑓2 −
) ) ; 𝑥, 𝑦󵄩󵄩󵄩󵄩
1+𝑦
󵄩󵄩
󵄩󵄩
󵄩 󵄩
= (𝛿𝑟,𝑥 , 𝛿𝑟,𝑦 ) 󵄩󵄩󵄩𝑔󵄩󵄩󵄩𝐶𝐵 (𝐾).
(69)
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