A Generalization of Lacunary Equistatistical Convergence of Positive Linear Operators

Abstract

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 𝐻_𝑤2 spaces, the spaces of all real valued continuous functions 𝑓 de…ned 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 𝐻_𝑤2 spaces. Then using the modulus of continuity, we study rates of lacunary equi-statistically convergence in 𝐻_𝑤2.

We recall here the concepts of equi-statistical convergence and lacunary equi-statistical convergence.

Let 𝑓 and 𝑓_𝑟 belong to 𝐶(𝑋), which is the space of all continuous real valued functions on a compact subset 𝑋 of the real numbers. {𝑓_𝑟} is said to be equi-statistically convergent to 𝑓 on 𝑋 and denoted by 𝑓_𝑟↠𝑓(equistat) if for every 𝜀>0, the sequence of real valued functions 𝑝_𝑟,𝜀(𝑥)1∶=𝑟𝑚≤𝑟∶𝑓_𝑚(𝑥)−𝑓(𝑥)≥𝜀 (1) converges uniformly to the zero function on 𝑋, which means that lim_𝑟→∞𝑝_𝑟,𝜀(⋅)_𝐶(𝑋)=0. (2) A lacunary sequence 𝜃={𝑘_𝑟} is an integer sequence such that 𝑘₀=0,ℎ_𝑟=𝑘_𝑟−𝑘_𝑟−1⟶∞as𝑟⟶∞. (3)

In this paper the intervals determined by 𝜃 will be denoted by 𝐼_𝑟=(𝑘_𝑟−1,𝑘_𝑟], and the ratio 𝑘_𝑟/𝑘_𝑟−1 will be abbreviated by 𝑞_𝑟.

Let 𝜃 be a lacunary sequence then {𝑓_𝑟}_𝑟∈ℕ is said to be lacunary equi-statistically convergent to 𝑓 on 𝑋 and denoted by 𝑓_𝑟↠𝑓(𝜃-equistat) if for every 𝜀>0, the sequence of real valued functions {𝑠_𝑟,𝜀}_𝑛∈ℕ defined by 𝑠_𝑟,𝜀(𝑥)1∶=ℎ_𝑟𝑚∈𝐼_𝑟∶𝑓_𝑚(𝑥)−𝑓(𝑥)≥𝜀 (4) uniformly converges to zero function on 𝑋, which means that lim_𝑛→∞𝑠_𝑟,𝜀(⋅)_𝐶(𝑋)=0. (5)

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.

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_𝑟𝑡^𝑣,𝑥↠𝑥^𝜈,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡,𝑣=0,1,2, (6) then for all 𝑓∈𝐶(𝑋), 𝐿_𝑟𝑓,𝑥↠𝑓,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡. (7)

We turn to introducing some notation and the basic definitions used in this paper. Throughout this paper 𝐼=[0,∞). Let 𝐶(𝐼)∶=𝑓∶𝑓isareal-valuedcontinuousfunctionon𝐼, (8) and 𝐶_𝐵(𝐼)∶=𝑓∈𝐶(𝐼)∶𝑓isboundedfunctionon𝐼. (9) Consider the space 𝐻_𝑤 of all real-valued functions 𝑓 defined on 𝐼 and satisfying 𝑓(𝑥)𝑦𝑥−𝑓≤𝑤𝑓;+𝑦1+𝑥,1+𝑦 (10) where 𝑤 is the modulus of continuity defined by 𝑤𝑓;𝛿∶=sup_{𝑥,𝑦∈𝐼𝑥−𝑦≤𝛿}𝑓(𝑥)−𝑓𝑦,forany𝛿>0 (11) (see [9]). Let 𝐾∶=𝐼²=[0,∞)×[0,∞), then the norm on 𝐶_𝐵(𝐾) is given by 𝑓∶=sup_{𝑥,𝑦∈𝐾}𝑓𝑥,𝑦,𝑓∈𝐶_𝐵(𝐾), (12) 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, 𝛿^′₁, 𝛿^′₂, 𝛿₁^′′, and 𝛿₂^′′>0,(i)𝑤₂(𝑓;𝛿₁,⋅) and 𝑤₂(𝑓;⋅,𝛿₂) are nonnegative increasing functions on [0,∞),(ii)𝑤₂(𝑓;𝛿^′₁+𝛿₁^′′,𝛿₂)≤𝑤₂(𝑓;𝛿^′₁,𝛿₂)+𝑤₂(𝑓;𝛿₁^′′,𝛿₂), (iii)𝑤₂(𝑓;𝛿₁,𝛿^′₂+𝛿₂^′′)≤𝑤₂(𝑓;𝛿₁,𝛿^′₂)+𝑤₂(𝑓;𝛿₁,𝛿₂^′′), (iv)lim_{𝛿1,𝛿2→0}𝑤₂(𝑓;𝛿₁,𝛿₂)=0.

The space 𝐻_𝑤2 is of all real-valued functions 𝑓 defined on 𝐾 and satisfying 𝑓(𝑢,𝑣)−𝑓𝑥,𝑦≤𝑤₂𝑢𝑓;−𝑥1+𝑢,𝑣1+𝑥−𝑦1+𝑣.1+𝑦 (13) It is clear that any function in 𝐻_𝑤2 is continuous and bounded on 𝐾.

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(𝐼^𝑚), the space of all continuous real valued functions on the subset 𝐼^𝑚 of R^𝑚, and the real 𝑚-dimensional space. We first consider the case of 𝑚=2. Following [7] we can state the following theorem.

Theorem 2. Let 𝜃={𝑘_𝑟} be a lacunary sequence, and let {𝐿_𝑟} be a sequence of positive linear operators from 𝐻_𝑤2 into 𝐶_𝐵(𝐾). 𝐿_𝑟 is satisfying 𝐿_𝑟(𝑓_𝑣;𝑥,𝑦)↠𝑓_𝑣(𝑥,𝑦)(𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡), 𝑣=0,1,2, where 𝑓_𝑘(𝑢,𝑣)∈𝐻_𝑤2, 𝑘=0,1,2,3, 𝑓₀()𝑓𝑢,𝑣=1,₁()=𝑢𝑢,𝑣,𝑓𝑢+1₂()=𝑣𝑢,𝑣,𝑓𝑣+1₃()=𝑢𝑢,𝑣𝑢+1²+𝑣𝑣+1², (14) then for all 𝑓∈𝐻_𝑤2, 𝐿_𝑟𝑓;𝑥,𝑦↠𝑓𝑥,𝑦,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡. (15)

Proof. Let (𝑥,𝑦)∈𝐾 be a fixed point, 𝑓∈𝐻_𝑤2, 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−𝑥𝑢+1𝑥+1<𝛿₁,𝑣−𝑦𝑣+1𝑦+1<𝛿₂. (16) Let 𝐾_𝛿1,𝛿2)𝑢∶=(𝑢,𝑣∈𝐾∶−𝑥1+𝑢1+𝑥<𝛿₁,𝑣−𝑦𝑣+1𝑦+1<𝛿₂. (17) Hence, 𝑓(𝑢,𝑣)−𝑓𝑥,𝑦=𝑓(𝑢,𝑣)−𝑓𝑥,𝑦_{𝜒𝐾𝛿12,𝛿(𝑢,𝑣)}+𝑓(𝑢,𝑣)−𝑓𝑥,𝑦_{𝜒𝛿12𝐾⧵𝐾,𝛿(𝑢,𝑣)}<𝜀+2𝑀_{𝜒𝛿12𝐾⧵𝐾,𝛿(𝑢,𝑣)}, (18) where 𝜒_𝑃 denotes the characteristic function of the set 𝑃. Observe that 𝜒_{𝐾⧵𝐾𝛿12,𝛿}()≤1𝑢,𝑣𝛿²₁𝑢−𝑥1+𝑢1+𝑥²+1𝛿²₂𝑣−𝑦𝑣+1𝑦+1². (19) Using (18), (19), and 𝑀∶=‖𝑓‖ we have 𝑓(𝑢,𝑣)−𝑓𝑥,𝑦≤𝜀+2𝑀𝛿²𝑢−𝑥1+𝑢1+𝑥²+𝑣−𝑦𝑣+1𝑦+1², (20) where 𝛿∶=min{𝛿₁,𝛿₂}.
By the linearity and positivity of the operators {𝐿_𝑟} and by (18), we have 𝐿_𝑟𝑓₁−𝑢𝑓1+𝑢⁰²+𝑓₂−𝑣𝑓1+𝑣⁰²;𝑥,𝑦≤𝐿_𝑟𝑓₃𝑥;𝑥,𝑦−2𝐿1+𝑥^𝑟𝑓₁+𝑦;𝑥,𝑦𝐿1+𝑦^𝑟𝑓₂;𝑥,𝑦𝐿_𝑟𝑓₃𝑥;𝑥,𝑦−2𝐿1+𝑥^𝑟𝑓₁+𝑦;𝑥,𝑦𝐿1+𝑦^𝑟𝑓₂+𝑥;𝑥,𝑦1+𝑥²+𝑦1+𝑦²𝐿_𝑟𝑓₀;𝑥,𝑦. (21) Hence, we get 𝐿_𝑟𝑓;𝑥,𝑦−𝑓𝑥,𝑦≤𝐿_𝑟𝑓()+𝑓𝐿𝑢,𝑣−𝑓𝑥,𝑦;𝑥,𝑦𝑥,𝑦_𝑟𝑓₀;𝑥,𝑦−𝑓₀𝑥,𝑦≤𝐿_𝑟𝜀+2𝑀𝛿²𝑢−𝑥1+𝑢1+𝑥²+𝑣−𝑦𝑣+11+𝑦²𝐿;𝑥,𝑦+𝑀_𝑟𝑓₀;𝑥,𝑦−𝑓₀𝑥,𝑦≤𝐿_𝑟𝜀+2𝑀𝛿²𝑓₁−𝑥1+𝑥⋅𝑓₀²+𝑓₂−𝑦1+𝑦⋅𝑓₀²𝐿;𝑥,𝑦+𝑀_𝑟𝑓₀;𝑥,𝑦−𝑓₀𝑥,𝑦=𝐿_𝑟𝜀;𝑥,𝑦+𝐿_𝑟2𝑀𝛿²𝑓₁−𝑥𝑓1+𝑥⁰²+𝑓₂−𝑦𝑓1+𝑦⁰²𝐿;𝑥,𝑦+𝑀_𝑟𝑓₀;𝑥,𝑦−𝑓₀𝑥,𝑦≤𝜀+2𝑀𝛿²𝐿_𝑟𝑓₃;𝑥,𝑦−𝑓₃+𝑥,𝑦4𝑀𝛿²𝐿_𝑟𝑓₂;𝑥,𝑦−𝑓₂+𝑥,𝑦4𝑀𝛿²𝐿_𝑟𝑓₁;𝑥,𝑦−𝑓₁+𝑥,𝑦𝜀+𝑀+4𝑀𝛿²𝐿_𝑟𝑓₀;𝑥,𝑦−𝑓₀=𝑥,𝑦2𝑀𝛿²𝐿_𝑟𝑓₃;𝑥,𝑦−𝑓₃+𝑥,𝑦4𝑀𝛿²𝐿_𝑟𝑓₂;𝑥,𝑦−𝑓₂+𝑥,𝑦4𝑀𝛿²𝐿_𝑟𝑓₁;𝑥,𝑦−𝑓₁𝐿𝑥,𝑦+𝜀+𝑁_𝑟𝑓₀;𝑥,𝑦−𝑓₀𝑥,𝑦, (22) where 𝑁∶=𝜀+𝑀+4𝑀/𝛿². For a given 𝜇>0, choose 𝜀>0 such that 𝜀<𝜇. Define the following sets:𝐷_𝜇𝑥,𝑦∶=𝑚∈ℕ∶𝐿_𝑚,𝐷𝑓;𝑥,𝑦−𝑓𝑥,𝑦≥𝜇_𝑣𝜇𝑥,𝑦∶=𝑚∈ℕ∶𝐿_𝑚𝑓;𝑥,𝑦−𝑓𝑥,𝑦≥𝜇−𝜀,4𝑁 (23) where  𝑣=0,1,2,3. Then from (22) we clearly have 𝐷_𝜇𝑥,𝑦⊆³_𝑣=0𝐷^𝑣_𝜇𝑥,𝑦. (24) Therefore define the following real valued functions: 𝑠_𝑟,𝜇1𝑥,𝑦∶=ℎ_𝑟𝑚∈𝐼_𝑟∶𝐿_𝑚,𝑠𝑓;𝑥,𝑦−𝑓𝑥,𝑦≥𝜇_{𝑣𝑟,𝜇}1𝑥,𝑦∶=ℎ_𝑟𝑚∈𝐼_𝑟∶𝐿_𝑚𝑓_𝑣;𝑥,𝑦−𝑓𝑥,𝑦≥𝜇−𝜀,4𝑁 (25) where 𝑣=0,1,2,3. Then by the monotonicity and (24) we get 𝑠_𝑟,𝜇𝑥,𝑦≤³_𝑣=0𝑠^𝑣_𝑟,𝜇𝑥,𝑦 (26) for all 𝑥∈𝑋, and this implies the inequality 𝑠_𝑟,𝜇(⋅)_𝐾≤³_𝑣=0𝑠^𝑣_𝑟,𝜇(⋅)_𝐾. (27) Taking limit in (27) as 𝑟→∞ and using (14) we have lim_𝑟→∞𝑠_𝑟,𝜇(⋅)_𝐾=0. (28) Then for all 𝑓∈𝐻_𝑤2, we conclude that 𝐿_𝑟𝑓;𝑥,𝑦↠𝑓𝑥,𝑦,𝜃-equistat. (29)

Now replace 𝐼² by 𝐼^𝑚=[0,∞)×⋯×[0,∞) and by an induction, we consider the modulus of continuity type function 𝑤_𝑚 as in the function 𝑤₂. Then let 𝐻_𝑤𝑚 be the space of all real-valued functions 𝑓 satisfying 𝑓𝑢₁,𝑢₂,…,𝑢_𝑚−𝑓𝑥₁,𝑥₂,…,𝑥_𝑚≤𝑤₂𝑢𝑓;¹𝑢₁−𝑥+1𝑥¹₁𝑢+1,…,^𝑚𝑢_𝑚−𝑥+1𝑥^𝑚_𝑚.+1 (30)

Therefore, using a similar technique in the proof of Lemma 1 one can obtain the following result immediately.

Theorem 3. Let 𝜃={𝑘_𝑟} be a lacunary sequence, and let {𝐿_𝑟} be a sequence of positive linear operators from 𝐻_𝑤𝑚 into 𝐶_𝐵(𝐼^𝑚). 𝐿_𝑟 is satisfying 𝐿_𝑟𝑓_𝑣;𝑥,𝑦↠𝑓_𝑣𝑥,𝑦,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡,𝑣=0,1,2,…,𝑚+1, (31) where 𝑓_𝑘(𝑢₁,𝑢₂,…,𝑢_𝑚)∈𝐻_𝑤𝑚, 𝑘=0,1,2,…𝑚+1, 𝑓₀𝑢₁,𝑢₂,…,𝑢_𝑚𝑓=1,₁𝑢₁,𝑢₂,…,𝑢_𝑚=𝑢₁𝑢₁,⋮𝑓+1_𝑚𝑢₁,𝑢₂,…,𝑢_𝑚=𝑢_𝑚𝑢_𝑚,𝑓+1_𝑚+1𝑢₁,𝑢₂,…,𝑢_𝑚=𝑢₁𝑢₁+1²+𝑢₂𝑢₂+1²𝑢+⋯+^𝑚𝑢_𝑚+1². (32) Then for all 𝑓∈𝐻_𝑤𝑚, 𝐿_𝑟𝑓;𝑢₁,𝑢₂,…,𝑢_𝑚↠𝑓𝑢₁,𝑢₂,…,𝑢_𝑚,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡. (33) Assume that 𝐼=[0,∞), 𝐾∶=𝐼×𝐼. One considers the following positive linear operators defined on 𝐻_𝑤2(𝐾): 𝐵_𝑛=1𝑓;𝑥,𝑦(1+𝑥)^𝑛1+𝑦^𝑛×^𝑛𝑛_𝑘=0𝑙=0𝑓𝑘,𝑙𝑛−𝑘+1𝑛𝑘𝑛𝑙𝑥𝑛−𝑙+1^𝑘𝑦^𝑙, (34) where 𝑓∈𝐻_𝑤2, (𝑥,𝑦)∈𝐾 and 𝑛∈ℕ.

Lemma 4. Let 𝜃={𝑘_𝑟} be a lacunary sequence, and let 𝐵_𝑛=1𝑓;𝑥,𝑦(1+𝑥)^𝑛1+𝑦^𝑛×^𝑛𝑛_𝑘=0𝑙=0𝑓𝑘,𝑙𝑛−𝑘+1𝑛𝑘𝑛𝑙𝑥𝑛−𝑙+1^𝑘𝑦^𝑙 (35) be a sequence of positive linear operators from 𝐻_𝑤2 into 𝐶_𝐵(𝐾). If 𝐵_𝑛 is satisfying 𝐵_𝑛𝑓_𝑣;𝑥,𝑦↠𝑓_𝑣,𝑓𝑥,𝑦𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡,𝑣=0,1,2,3,₀()𝑓𝑢,𝑣=1,₁()=𝑢𝑢,𝑣,𝑓𝑢+1₂()=𝜈𝑢,𝑣,𝑓𝜈+1₃()=𝑢𝑢,𝑣𝑢+1²+𝜈𝜈+1², (36) then for all 𝑓∈𝐻_𝑤2, 𝐵_𝑛𝑓;𝑥,𝑦↠𝑓𝑥,𝑦,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡. (37)

Proof. Assume that (36) holds, and let 𝑓∈𝐻_𝑤2. Since (1+𝑥)^𝑛=^𝑛_𝑘=0𝑛𝑘𝑥^𝑘,1+𝑦^𝑛=^𝑛_𝑙=0𝑛𝑙𝑦^𝑙, (38) it is clear that, for all 𝑛∈ℕ, 𝐵_𝑛𝑓₀=1;𝑥,𝑦(1+𝑥)^𝑛1+𝑦^𝑛𝑛𝑛_𝑘=0𝑙=0𝑛𝑘𝑥^𝑘𝑛𝑙𝑦^𝑘=1(1+𝑥)^𝑛1+𝑦^𝑛𝑛_𝑘=0𝑛𝑘𝑥^𝑘𝑛_𝑙=0𝑛𝑙𝑦^𝑘=1. (39) Now, by assumption we have 𝐵_𝑛𝑓₀;𝑥,𝑦↠𝑓₀𝑥,𝑦,𝜃-equistat. (40) Using the definition of 𝐵_𝑛, we get 𝐵_𝑛𝑓₁=1;𝑥,𝑦(1+𝑥)^𝑛1+𝑦^𝑛×^𝑛_𝑘=1𝑘╱(𝑛−𝑘+1)()𝑛𝑘𝑥𝑘╱𝑛−𝑘+1+1^𝑘𝑛_𝑙=0𝑛𝑙𝑦^𝑙=1(1+𝑥)^𝑛1+𝑦^𝑛𝑛_𝑘=1𝑘𝑛𝑘𝑥𝑛+1^𝑘𝑛_𝑙=0𝑛𝑙𝑦^𝑙=1(1+𝑥)^𝑛𝑛−1_𝑘=0𝑘+1𝑛𝑥𝑛+1𝑘+1^𝑘+1=𝑥(1+𝑥)^𝑛𝑛−1_𝑘=0𝑘+1𝑛𝑥𝑛+1𝑘+1^𝑘. (41) Since 𝑛=𝑘𝑛𝑘+1𝑛−1𝑘+1 (42) we get 𝐵_𝑛𝑓₁=𝑥;𝑥,𝑦(1+𝑥)^𝑛𝑛−1_𝑘=0𝑘+1𝑛𝑛+1𝑘𝑥𝑘+1𝑛−1_𝑘=𝑥(1+𝑥)^𝑛𝑛−1_𝑘=0𝑛𝑘𝑥𝑛+1𝑛−1_𝑘=𝑥(1+𝑥)(1+𝑥)^𝑛−1⋅𝑛⋅𝑛+1^𝑛−1_𝑘=0𝑘𝑥𝑛−1_𝑘=𝑛𝑥𝑛+1.𝑥+1 (43) So, we have 𝐵_𝑛𝑓₁;𝑥,𝑦−𝑓₁=𝑥𝑥,𝑦𝑛𝑥+1.𝑛+1−1 (44) The fact that lim_𝑛→∞(𝑛/(𝑛+1))=1 and using a similar technique as in the proof of Lemma 1, we get lim_𝑟→∞1ℎ_𝑟𝑚∈𝐼_𝑟∶𝐵_𝑛𝑓₁;𝑥,𝑦−𝑓₁𝑥,𝑦≥𝜀=0. (45) Hence we have 𝐵_𝑛𝑓₂;𝑥,𝑦↠𝑓₂𝑥,𝑦,𝜃-equistat. (46) Also we have 𝐵_𝑛𝑓₃;𝑥,𝑦↠𝑓₃𝑥,𝑦,𝜃-equistat. (47) To see this, by the definition of 𝐵_𝑛, we first write𝐵_𝑛𝑓₃=1;𝑥,𝑦(1+𝑥)^𝑛1+𝑦^𝑛𝑛𝑛_𝑘=0𝑙=0𝑛𝑘𝑛𝑙𝑥^𝑘𝑦^𝑙×𝑘²(𝑛+1)²+𝑙²(𝑛+1)²=1(1+𝑥)^𝑛1+𝑦^𝑛𝑛_𝑘=1𝑘²(𝑛+1)²𝑛𝑘𝑥^𝑘𝑛_𝑙=0𝑛𝑙𝑦^𝑙+1(1+𝑥)^𝑛1+𝑦^𝑛𝑛_𝑘=0𝑛𝑘𝑥^𝑘∞_𝑙=1𝑙²(𝑛+1)²𝑛𝑙𝑦^𝑙=1(1+𝑥)^𝑛𝑛_𝑘=1𝑘(𝑛+1)²𝑥^𝑘+1(1+𝑥)^𝑛𝑛_𝑘=2𝑘(𝑘−1)(𝑛+1)²𝑛𝑘𝑥^𝑘+11+𝑦^𝑛𝑛_𝑙=2𝑙(𝑙−1)(𝑛+1)²𝑛𝑙𝑦^𝑙+11+𝑦^𝑛𝑛_𝑙=1𝑙(𝑛+1)²𝑦^𝑙. (48) Then, 𝐵_𝑛𝑓₃=𝑛();𝑥,𝑦𝑛−1(𝑛+1)²𝑥²(𝑥+1)²+𝑛(𝑛+1)²𝑥()+𝑛𝑥+1(𝑛+1)²𝑦+𝑦𝑦+1₂𝑦+1²𝑛(𝑛−1)(𝑛+1)² (49) which implies that 𝐵_𝑛𝑓₃;𝑥,𝑦−𝑓₃=𝑥𝑥,𝑦₂(𝑥+1)²+𝑦²𝑦+1²𝑛(𝑛−1)(𝑛+1)²+𝑛−1(𝑛+1)²𝑥+𝑦𝑥+1.𝑦+1 (50) Since lim_𝑛→∞𝑛(𝑛−1)(𝑛+1)²=1,lim_𝑛→∞𝑛(𝑛+1)²=0, (51) we get lim_𝑟→∞1ℎ_𝑟𝑚∈𝐼_𝑟∶𝐵_𝑛𝑓₃;𝑥,𝑦−𝑓₃𝑥,𝑦≥𝜀=0. (52) Thus 𝐵_𝑛(𝑓₃;𝑥,𝑦)↠𝑓₃(𝑥,𝑦),(𝜃-equistat). Therefore we obtain that for all 𝑓∈𝐻_𝑤2, 𝐵_𝑛(𝑓;𝑥,𝑦)↠𝑓(𝑥,𝑦),(𝜃-equistat).

3. Rates of Lacunary Equistatistical Convergence

In this section we study the order of lacunary equi-statistical convergence of a sequence of positive linear operators acting on 𝐻_𝑤2(𝐾), where 𝐾=𝐼^𝑚. To achieve this we first consider the case of 𝑚=2.

Definition 5. A sequence {𝑓_𝑟} is called lacunary equi-statistically convergent to a function 𝑓 with rate 0<𝛽<1 if for every 𝜀>0, lim_𝑟→∞𝑠_𝑟,𝜀𝑥,𝑦𝑟^−𝛽=0, (53) where 𝑠_𝑟,𝜀(𝑥,𝑦) is given in Lemma 1. In this case it is denoted by 𝑓_𝑟−𝑓=𝑜𝑟^−𝛽,𝜃-equistaton𝐾=𝐼×𝐼. (54)

Lemma 6. Let {𝑓_𝑟} and {𝑔_𝑟} be two sequences of functions in 𝐻_𝑤2(𝐾), with 𝑓_𝑟−𝑓=𝑜𝑟^−𝛽1,,𝑔𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡_𝑟−𝑔=𝑜𝑟^−𝛽2,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡. (55) Then one has 𝑓_𝑟+𝑔_𝑟−𝑓+𝑔=𝑜𝑟^−𝛽,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡, (56) where 𝛽=min{𝛽₁,𝛽₂}.

Proof. Assume that 𝑓_𝑟−𝑓=𝑜(𝑟^−𝛽1), (𝜃-equistat) and 𝑔_𝑟−𝑔=𝑜(𝑟^−𝛽2), (𝜃-equistat) on 𝐾. For all 𝜀>0, consider the following functions: 𝑠_𝑟,𝜀1𝑥,𝑦∶=ℎ_𝑟𝑛∈𝐼_𝑟∶𝑓_𝑛+𝑔_𝑛−,𝑠𝑥,𝑦𝑓+𝑔𝑥,𝑦≥𝜀_1𝑟,𝜀1𝑥,𝑦∶=ℎ_𝑟𝑛∈𝐼_𝑟∶𝑓_𝑛(𝑥)(𝑥≥𝜀−𝑓)2,𝑠²_𝑟,𝜀1𝑥,𝑦∶=ℎ_𝑟𝑛∈𝐼_𝑟∶𝑔_𝑛(𝑥)(𝑥≥𝜀−𝑔)2. (57) Then we have lim_𝑟→∞𝑠_𝑟,𝜀𝑥,𝑦_{𝐻𝑤2(𝐾)}𝑟^−𝛽=lim_𝑟→∞𝑠¹_𝑟,𝜀𝑥,𝑦𝑟^−𝛽+lim_𝑟→∞𝑠²_𝑟,𝜀𝑥,𝑦𝑟^−𝛽,𝑠_𝑟,𝜀𝑥,𝑦𝑟^−𝛽≤𝑠¹_𝑟,𝜀𝑥,𝑦𝑟^−𝛽+𝑠²_𝑟,𝜀𝑥,𝑦𝑟^−𝛽≤𝑠¹_𝑟,𝜀𝑥,𝑦𝑟^−𝛽1+𝑠²_𝑟,𝜀𝑥,𝑦𝑟^−𝛽2, (58) and hence 𝑠_𝑟,𝜀𝑥,𝑦_{𝐻𝑤2(𝐾)}𝑟^−𝛽≤𝑠¹_𝑟,𝜀𝑥,𝑦_{𝐻𝑤2(𝐾)}𝑟^−𝛽1+𝑠²_𝑟,𝜀𝑥,𝑦_{𝐻𝑤2(𝐾)}𝑟^−𝛽2. (59) Taking limit as 𝑟→∞ and using the assumption complete the proof.

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∶𝐻_𝑤2(𝐾)→𝐻_𝑤2(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;𝛿_𝑟,𝑥,𝛿_𝑟,𝑦)=𝑜(𝑟^−𝛽2),(𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡) on 𝐾 with 𝛿_𝑟,𝑥=𝐿_𝑟𝑢−𝑥1+𝑢1+𝑥²,𝛿,𝑥_𝑟,𝑦=𝐿_𝑟𝑣−𝑦1+𝑣1+𝑦²,𝑦. (60)Then 𝐿_𝑟𝑓;𝑥,𝑦−𝑓𝑥,𝑦=𝑜𝑟^−𝛽,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡𝑜𝑛𝐾, (61) where 𝛽=min{𝛽₁,𝛽₂}.

Proof. Let 𝑓∈𝐻_𝑤2(𝐾) and 𝑥∈𝐾. Use 𝐿_𝑟𝑓;𝑥,𝑦−𝑓𝑥,𝑦≤𝐿_𝑟𝑓()+𝑓𝐿𝑢,𝑣−𝑓𝑥,𝑦;𝑥,𝑦𝑥,𝑦_𝑟𝑓₀;𝑥,𝑦−𝑓𝑥,𝑦≤𝐿_𝑟𝑤₂𝑢𝑓;−𝑥1+𝑢,𝑣1+𝑥−𝑦1+𝑣+𝑓𝐿1+𝑦;𝑥,𝑦𝑥,𝑦_𝑟𝑓₀;𝑥,𝑦−𝑓₀≤𝑥,𝑦1+𝐿_𝑟𝑓₀;𝑥,𝑦𝑤₂𝑓;𝛿_𝑟,𝑥,𝛿_𝑟,𝑦+𝑀𝐿_𝑟𝑓₀;𝑥,𝑦−𝑓₀𝑥,𝑦=2𝑤₂𝑓;𝛿_𝑟,𝑥,𝛿_𝑟,𝑦+𝑀𝐿_𝑟𝑓₀;𝑥,𝑦−𝑓₀𝑥,𝑦+𝑤₂𝑓;𝛿_𝑟,𝑥,𝛿_𝑟,𝑦𝐿_𝑟𝑓₀;𝑥,𝑦−𝑓₀𝑥,𝑦, (62) where 𝑀=‖𝑓‖_{𝐻𝑤2(𝐾)}. Using inequality (62), conditions (i) and (ii) we get lim_𝑟→∞1𝑟^−𝛽1ℎ_𝑟𝑟∈𝐼_𝑟∶𝐿_𝑟𝑓;𝑥,𝑦−𝑓𝑥,𝑦≥𝜀=0, (63) so we have 𝐿_𝑟𝑓;𝑥,𝑦−𝑓𝑥,𝑦=𝑜𝑟^−𝛽,𝜃-equistaton𝐾. (64)

Finally we give the rate of lacunary equi-statistical convergence for the operators 𝐿_𝑟(𝑓,𝑥) by using the Peetre's 𝐾-functional in the space 𝐻_𝑤2(𝐾). The Peetre 𝐾-functional of function 𝑓∈𝐻_𝑤2(𝐾) is defined by 𝐾𝑓;𝛿_𝑟,𝑥,𝛿_𝑟,𝑦=inf_{𝑔∈𝐻𝑤2(𝐾)}𝑓−𝑔_{𝐶𝐵(𝐾)}+𝛿𝑔_{𝐶𝐵(𝐾)}, (65) where 𝑓_{𝐶𝐵(𝐾)}=sup_{𝑥,𝑦∈𝐾}𝑓𝑥,𝑦. (66)

Theorem 8. Let 𝑓∈𝐻_𝑤2(𝐾) and {𝐾(𝑓;𝛿_𝑟,𝑥,𝛿_𝑟,𝑦)} be the sequence of Peetre's K-functional. If 𝛿_𝑟,𝑥=𝐿_𝑟𝑓₁−𝑥1+𝑥;𝑥,𝑦_{𝐶𝐵(𝐾)}+𝐿_𝑟𝑓₁−𝑥1+𝑥²;𝑥,𝑦_{𝐶𝐵(𝐾)}𝑔_{𝐶𝐵(𝐾)},𝛿_𝑟,𝑦=𝐿_𝑟𝑓₂−𝑦1+𝑦;𝑥,𝑦_{𝐶𝐵(𝐾)}+𝐿_𝑟𝑓₂−𝑦1+𝑦²;𝑥,𝑦_{𝐶𝐵(𝐾)}𝑔_{𝐶𝐵(𝐾)},lim_𝑟→∞𝛿_𝑟,𝑥=0,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡lim_𝑟→∞𝛿_𝑟,𝑦=0,𝜃-𝑒𝑞𝑢𝑖𝑠𝑡𝑎𝑡 (67) on 𝑥,𝑦∈𝐾, then 𝐿_𝑟𝑓;𝑥,𝑦−𝑓𝑥,𝑦_{𝐶𝐵(𝐾)}≤𝐾𝑓;𝛿_𝑟,𝑥,𝛿_𝑟,𝑦. (68)

Proof. For each 𝑔∈𝐻_𝑤2(𝐾), we get 𝐿_𝑟𝑔;𝑥,𝑦−𝑔𝑥,𝑦_{𝐶𝐵(𝐾)}≤𝑔_{𝐶𝐵(𝐾)}𝐿_𝑟𝑓₁−𝑥1+𝑥;𝑥,𝑦_{𝐶𝐵(𝐾)}+𝐿_𝑟𝑓₁−𝑥1+𝑥²+𝑔;𝑥,𝑦_{𝐶𝐵(𝐾)}𝐿_𝑟𝑓₂−𝑦1+𝑦;𝑥,𝑦_{𝐶𝐵(𝐾)}+𝐿_𝑟𝑓₂−𝑦1+𝑦²=𝛿;𝑥,𝑦_𝑟,𝑥,𝛿_𝑟,𝑦𝑔_{𝐶𝐵(𝐾).} (69)