ℐ-αβ-statistical relative uniform convergence for double sequences and its applications

Nilay Sahin Bayram; Sevda Yıldız

doi:10.1515/dema-2025-0163

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ℐ-αβ-statistical relative uniform convergence for double sequences and its applications

Nilay Sahin Bayram and Sevda Yıldız

Published/Copyright: October 4, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

This article introduces a novel concept of convergence, referred to as ℐ - α β -statistical relative uniform convergence, for double sequences of functions. This notion, which is proposed for the first time in this article, is explored in depth, leading to the establishment of a Korovkin-type approximation theorem within this framework. The illustrative example demonstrates that the proposed convergence is indeed stronger than previously known forms. Additionally, this article investigates the rate of ℐ 2 - α β -statistical relative uniform convergence, providing explicit computations to support the findings. The results contribute to the understanding of ideal statistical convergence and open up new perspectives for approximation theory.

Keywords: Korovkin-type theorem; positive linear operator; ideal relative convergence for double sequences; ℐ2-α β-statistical relative uniform convergence

MSC 2010: 40A35; 40B05; 41A36

1 Introduction

The concept of statistical convergence was initially introduced by Steinhaus [1] and independently by Fast [2] as a generalization of classical convergence for real sequences. Over time, various generalizations and applications of statistical convergence have been explored in different spaces, notably through the significant works of Fridy [3], Fridy and Orhan [4], and Šalát [5]. In recent years, the concept of statistical convergence has been further extended in several important directions, such as:

Lacunary statistical convergence introduced by Fridy and Orhan [4];
λ -statistical convergence, explored by Çolak and Bektaş [6];
Statistical relative uniform convergence introduced by Demirci and Orhan [7];
Statistical convergence using the power series method, pioneered by Ünver and Orhan [8], with subsequent developments in the objectives of this method by various authors in 2022 and 2023 [8–11];
Ideal convergence, a concept developed by Kostyrko et al. [12], as well as independently by Nuray and Ruckle [13].

In 2008, ideal convergence was extended to sequences of real functions and double sequences by Das et al. [14]. Subsequently, Dündar and Altay [15] investigated various ideal convergence notions for double sequences of real-valued functions. Building on these advances, Aktuğlu [16] introduced a new variant of statistical convergence, known as α β -statistical convergence of order γ , which has proven to be a significant extension of both classical and statistical convergences. This concept also encompasses several other statistical-type convergences discussed in this article.

Further developments in double sequences came from Altundağ and Sözbir [17], who introduced α β -statistical convergence and strong α β -summability for double sequences, exploring the interplay between these concepts. Savas and Das [18] later combined statistical convergence with ℐ -convergence, leading to the emergence of ℐ -statistical convergence, which was subsequently generalized to double sequences by Belen and Yildirim [19]. In a more recent advancement, Ghosal and Mandal [20] introduced ℐ - α β -statistical convergence for single sequences, which further generalizes the notion of ℐ -statistical convergence. Then, ℐ - α β -statistical convergence for double sequences came from Yıldız and Šahin Bayram [21].

Based on the research mentioned earlier, this article introduces a novel type of convergence called ℐ - α β -statistical relative uniform convergence for double sequences of functions, a concept presented for the first time. Then, the approximation theorem of the Korovkin-type via this new type of convergence has been proved. An example is provided to illustrate that this new convergence is indeed stronger than previous types. Finally, the rate of ℐ 2 - α β -statistical relative uniform convergence is calculated, contributing to a deeper understanding of this new convergence concept.

2 Preliminaries

For our investigation, the following is requisite

2.1 Statistical-type convergences

First, recall the main concept of convergence methods for double sequences.

Definition 1

[22] A double sequence x = ( x i j ) is said to be convergent in Pringsheim’s sense ( P -convergent) if

∀ ε > 0 , ∃ J = J ( ε ) , ∀ i , j > J , ∣ x i j − L ∣ < ε .

This limit is represented as P − lim i , j x i j = L .

Definition 2

[22] A double sequence x = ( x i j ) is called bounded if

∃ K > 0 , ∀ ( i , j ) ∈ N 2 , ∣ x i j ∣ ≤ K , i.e.,

if ∥ x ∥ ( ∞ , 2 ) = sup i , j ∣ x i j ∣ < ∞ .

Definition 3

[23] Consider the double sequence x = ( x i j ) . It is said to be statistically convergent to the limit L if for every ε > 0 , the following holds:

P − lim m , n → ∞ ∣ { ( i , j ) : i ≤ m , j ≤ n , ∣ x i j − L ∣ ≥ ε } ∣ m n = 0 .

In this setting, the convergence is expressed as s t 2 − lim i , j x i j = L .

Now, let α ( n ) and β ( n ) be two sequences of positive numbers satisfying the following conditions:

C 1 : α , β are both non-decreasing, C 2 : β ( n ) ≥ α ( n ) , C 3 : β ( n ) − α ( n ) → ∞ , as n → ∞ ,

and let Λ denote the set of pairs ( α , β ) satisfying C 1 , C 2 , and C 3 [16].

Throughout this article, let ( α 1 , β 1 ) , ( α 2 , β 2 ) ∈ Λ .

Definition 4

[17] A double sequence x = ( x i j ) is said to be α β -statistically convergent to L , if for every ε > 0 ,

P − lim m , n 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ { ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : ∣ x i j − L ∣ ≥ ε } ∣ = 0 ,

which is denoted by s t 2 α β − lim i , j x i j = L , where D m α 1 β 1 and D n α 2 β 2 are closed intervals [ α 1 ( m ) , β 1 ( m ) ] and [ α 2 ( n ) , β 2 ( n ) ] , respectively.

Definition 5

[17] A double sequence x = ( x i j ) is said to be strongly α β -summable to L , if for every ε > 0 ,

P − lim m , n 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∑ i ∈ D m α 1 β 1 ∑ j ∈ D n α 2 β 2 ∣ x i j − L ∣ = 0 .

2.2 Ideal-type convergences

We begin by recalling some basic terminology and notation regarding ideals.

The notion of ℐ -convergence, introduced by Kostyrko et al. [12], generalizes statistical convergence through the use of an ideal ℐ . Let S be a non-empty set. A family ℐ of subsets of S is called an ideal on S if it satisfies the following properties:

∅ ∈ ℐ ,
if A 1 , A 2 ∈ ℐ , then A 1 ∪ A 2 ∈ ℐ ,
if A 1 ∈ ℐ and A 2 ⊂ A 1 , then A 2 ∈ ℐ .

An ideal ℐ is called admissible if { x } ∈ ℐ for every x ∈ S . Moreover, ℐ is said to be non-trivial if S ∉ ℐ and ℐ ≠ { ∅ } .

For such a non-trivial ideal ℐ , the associated filter is defined as

ℱ = { U ⊂ S : ( ∃ A 1 ∈ ℐ ) such that U = S \ A 1 } .

Now, consider a non-trivial ideal ℐ 2 on the set N 2 . This ideal is said to be strongly admissible if for every i ∈ N , the sets { i } × N and N × { i } are in ℐ 2 . It is evident that any strongly admissible ideal is also admissible.

Define the collection

ℐ 2 0 = { A 2 ⊂ N 2 : ( ∃ m ( A 2 ) ∈ N ) such that ( i , j ) ∉ A 2 whenever i , j ≥ m ( A 2 ) } .

This set forms a strongly admissible, non-trivial ideal on N 2 [14]. Furthermore, an ideal ℐ 2 is strongly admissible if and only if ℐ 2 0 ⊂ ℐ 2 .

Throughout the rest of this article, we shall assume that ℐ 2 is a non-trivial, strongly admissible ideal on N 2 .

Definition 6

[19] A double sequence x = ( x i j ) of real numbers is said to be ℐ 2 ‐ statistically convergent to a number L , if for each ε > 0 and η > 0 ,

( m , n ) ∈ N 2 : 1 m n ∣ { i ≤ n , j ≤ m : ∣ x i j − L ∣ ≥ ε } ∣ ≥ η ∈ ℐ 2 .

Yıldız and Šahin Bayram [21] introduced the following definition of ℐ 2 ‐ α β -statistical convergence and gave well-known three important results that are stated as follows:

Definition 7

A double sequence x = ( x i j ) is said to be ℐ 2 ‐ α β -statistically convergent to L , if for every ε > 0 ,

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ × ∣ { ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : ∣ x i j − L ∣ ≥ ε } ∣ ≥ η ∈ ℐ 2 .

In this case, we write s t ( ℐ 2 α β ) − lim i , j x i j = L .

(i) If we take α 1 ( m ) = 1 , β 1 ( m ) = m for all m ∈ N and α 2 ( n ) = 1 , β 2 ( n ) = n for all n ∈ N , then ℐ 2 ‐ α β -statistical convergence is reduced to ℐ 2 -statistical convergence introduced in Definition 6.

(ii) Let λ = ( λ m ) and μ = ( μ n ) be two non-decreasing sequences of positive numbers tending to ∞ such that

λ m + 1 ≤ λ m + 1 , λ 1 = 1 , μ n + 1 ≤ μ n + 1 , μ 1 = 1 .

Then, in the case of α 1 ( m ) = m − λ m + 1 , β 1 ( m ) = m for all m ∈ N , and α 2 ( n ) = n − μ n + 1 , β 1 ( n ) = n for all n ∈ N , ℐ 2 − α β -statistical convergence is reduced to ℐ 2 − ( λ , μ ) -statistical convergence introduced in [19].

(iii) Recall that a double lacunary sequence θ r , s = { ( k r , l s ) } , which means there exist two increasing integers such that

k 0 = 0 , h r = k r − k r − 1 → ∞ , as r → ∞ , l 0 = 0 , h ¯ s = l s − l s − 1 → ∞ , as s → ∞ ,

If we take α 1 ( m ) = k m − 1 + 1 , β 1 ( m ) = k m for all m ∈ N and α 2 ( n ) = l n − 1 + 1 , β 1 ( n ) = l n for all n ∈ N , then ℐ 2 ‐ α β -statistical convergence of double sequence is reduced to ℐ 2 ‐ lacunary statistical convergence of double sequence introduced in [24].

The concept of uniform convergence of a sequence of functions relative to a scale function first given by Moore in [25]. Then, Chittenden [26] gave the definition that is equivalent to the definition given by Moore for single sequences. Demirci and Orhan [7] extended this idea to statistical convergence and gave approximation results, and then, Okçu Šahin and Dirik [27] introduced this type of convergence for double sequences as follows (see also [28]):

Definition 8

[27] The double function sequence ( f i j ) defined on any compact subset of the real two-dimensional space converges to the limit function f relatively uniformly if there exists a function σ ( x , y ) defined on any compact subset of the real two-dimensional space (which in the literature is called the scale function) such that for every ε > 0 , there is an integer n ε such that for m , n > n ε , the inequality

∣ f i , j ( x , y ) − f ( x , y ) ∣ < ε ∣ σ ( x , y ) ∣

holds uniformly in ( x , y ) . The double sequence ( f i j ) is said to converge uniformly relatively to the scale function σ , briefly, relatively uniformly convergent.

Ideal relative uniform convergence for double sequences of functions was introduced for the first time by Yıldız [29] in 2021. Then, in 2024, Yıldız and Šahin Bayram [21] introduced novel concept of ℐ 2 ‐ α β -statistical convergence. In view of these new-type of convergence methods in this article, we study an interesting type of convergence called ℐ 2 − α β -statistical relative uniform convergence.

3 Ideal α β -statistical-type convergences

Now, we introduce our new convergence methods. Let X 2 be a compact subset of R 2 .

Definition 9

A double sequence ( f i j ) is said to be ℐ 2 − α β -statistically pointwise convergent to f on X 2 , written in short as s t ( ℐ 2 α β ) − f i j → f for each ( x , y ) ∈ X 2 ,

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ × ∣ { ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : ∣ f i j ( x , y ) − f ( x , y ) ∣ ≥ ε } ∣ ≥ η ∈ ℐ 2 ,

written in short as s t ( ℐ 2 α β ) − f i j → f on X 2 .

Definition 10

A double sequence ( f i j ) is said to be ℐ 2 ‐ α β -statistically uniformly convergent to f on X 2 , written in short as s t ( ℐ 2 α β ) − f i j ⇉ f if for every ε > 0 ,

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ × ∣ { ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 ∣ f i j ( x , y ) − f ( x , y ) ∣ ≥ ε } ∣ ≥ η ∈ ℐ 2 .

Then, it is denoted s t ( ℐ 2 α β ) − f i j ⇉ f .

Definition 11

A double sequence ( f i j ) is said to be ℐ 2 ‐ α β -statistically relatively uniformly convergent to f on X 2 , if for every ε > 0 ,

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ × ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) ≥ ε ≥ η ∈ ℐ 2 .

In this case, we write s t ( ℐ 2 α β ) − f i j ⇉ f ( X 2 ; σ ) .

It is important to mention that choosing ℐ 2 = ℐ 2 0 leads to the notion of α β -statistical relative uniform convergence for double sequences – a recently introduced type of convergence. Moreover, if the scale function is substituted by a nonzero constant, this framework reduces to the classical α β -statistical convergence of sequences of functions [16,17].

Based on the preceding definition, we can directly state the following result.

Lemma 1

s t 2 α β − f i j ⇉ f on X 2 implies s t ( ℐ 2 α β ) − f i j ⇉ f on X 2 , which also implies s t ( ℐ 2 α β ) − f i j ⇉ f ( X 2 ; σ ) .

Nonetheless, the reverse implication of Lemma 1 does not hold in general, as demonstrated by the counterexample presented in the following.

Example 1

Let ℐ 2 = ℐ 2 ϕ , set of all subsets of N 2 with double natural density zero. For each ( i , j ) ∈ N 2 , let α 1 ( m ) = m 2 , β 1 ( m ) = ( m + 1 ) 2 , α 2 ( n ) = n 2 , β 2 ( n ) = ( n + 1 ) 2 and define h i j : [ 0 , 1 ] 2 → R by

h i j ( x , y ) = i j ; ( x , y ) ∉ 0 , 1 i × 0 , 1 j , ( i ∈ [ α 1 ( m ) , β 1 ( m ) ] , j ∈ [ α 2 ( n ) , β 2 ( n ) ] ∧ m = l 2 , n = k 2 , l , k ∈ N ) , i j ( 1 − i j x y ) ; ( x , y ) ∈ 0 , 1 i × 0 , 1 j , 0 ; otherwise.

Here, ( h i j ) is neither ℐ 2 − α β -statistically uniformly convergent nor α β -statistically uniformly convergent to the function h = 0 on the interval [ 0 , 1 ] 2 . In addition, ( h i j ) is not classically uniformly convergent to the function h = 0 . But it is seen to be s t ( ℐ 2 α β ) − h i j ⇉ h = 0 ( [ 0 , 1 ] 2 ; σ ) , with

σ ( x , y ) = 1 x 2 y 2 , ( x , y ) ∈ ( 0 , 1 ] × ( 0 , 1 ] , 1 , x = 0 or y = 0 .

Indeed, ∃ J = J ( ε ) ,

1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ S 2 h i j ( x , y ) σ ( x , y ) ≥ ε ≤ ( [ β 1 ( m ) − α 1 ( m ) ] + 1 ) ( [ β 2 ( n ) − α 2 ( n ) ] + 1 ) ( β 1 ( m ) − α 1 ( m ) + 1 ) ( β 2 ( n ) − α 2 ( n ) + 1 ) ; m = l 2 , n = k 2 , l , k ∈ N , [ J ] ( β 1 ( m ) − α 1 ( m ) + 1 ) ( β 2 ( n ) − α 2 ( n ) + 1 ) ; otherwise.

Example 2

Let ℐ 2 = ℐ 2 ϕ and A ∈ ℐ 2 ϕ . Let us take α 1 ( m ) = α 2 ( n ) = 1 , β 1 ( m ) = m , β 2 ( n ) = n and for each ( i , j ) ∈ N 2 , define h i j : [ 0 , 1 ] 2 → R by

(1) h i j ( x , y ) = i j , β 1 ( m ) − β 1 ( m ) − α 1 ( m ) + 1 + 1 ≤ i ≤ β 1 ( m ) , β 2 ( n ) − β 2 ( n ) − α 2 ( n ) + 1 + 1 ≤ j ≤ β 2 ( n ) ∧ ( m , n ) ∉ A i j , α 1 ( m ) ≤ i ≤ β 1 ( m ) , α 2 ( n ) ≤ j ≤ β 2 ( n ) ∧ ( m , n ) ∈ A i j x y 2 + i 2 j 2 x 2 y 2 , otherwise ,

and the scale function given by

(2) σ ( x , y ) = 1 x y , ( x , y ) ∈ ( 0 , 1 ] × ( 0 , 1 ] , 1 , x = 0 or y = 0 .

Then, for every ε > 0 ( 0 < ε < 1 ) , since

1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ S 2 h i j ( x , y ) σ ( x , y ) ≥ ε ≤ β 1 ( m ) − α 1 ( m ) + 1 β 2 ( n ) − α 2 ( n ) + 1 ( β 1 ( m ) − α 1 ( m ) + 1 ) ( β 2 ( n ) − α 2 ( n ) + 1 ) → 0 ,

as m , n → ∞ and ( m , n ) ∉ A . Hence,

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ × ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ S 2 h i j ( x , y ) σ ( x , y ) ≥ ε ≥ η ⊂ A ∪ { 1, 2 … , J 1 } ,

for some J 1 ∈ N . Clearly, the set on the right-hand side belongs to ℐ 2 ϕ and so the set on the left-hand side also belongs to ℐ 2 ϕ . This shows that s t ( ℐ 2 α β ) − h i j ⇉ h = 0 ( [ 0 , 1 ] 2 ; σ ) . However, ( h i j ) does not converge ℐ 2 -relatively uniformly to h .

Nevertheless, the sequence ( h i j ) fails to exhibit ℐ 2 - α β statistical uniform convergence, as well as α β statistical uniform convergence, toward the zero function on the domain [ 0 , 1 ] 2 . Furthermore, ( h i j ) does not converge uniformly in the classical sense to h = 0 over the same domain.

Definition 12

A double sequence ( f i j ) of functions defined on X 2 is said to be ℐ 2 − α β -relatively uniformly summable to f , if for every ε > 0 ,

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∑ i ∈ D m α 1 β 1 ∑ j ∈ D n α 2 β 2 sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) ≥ ε ∈ ℐ 2 .

Based upon the aforementioned definitions, we shall give the following special cases and some inclusion relations to show the effectiveness of newly proposed methods introduced in Definitions 11 and 12.

Let us take α 1 ( m ) = 1 , β 1 ( m ) = m for all m ∈ N and α 2 ( n ) = 1 , β 2 ( n ) = n for all n ∈ N ; then, ℐ 2 ‐ α β -statistical relative uniform convergence given in Definition 11, is reduced to ℐ 2 -statistical relative uniform convergence. If the case of the scale function is taken to be a constant different from zero, then we obtain ideal statistical convergence and ideal summability for function sequences (cf. [18,19,30,31]).
In the case of α 1 ( m ) = k m − 1 + 1 , β 1 ( m ) = k m for all m ∈ N and α 2 ( n ) = l n − 1 + 1 , β 1 ( n ) = l n for all n ∈ N , then, strong ℐ 2 − α β -relative uniform summability given in Definition 12 is reduced to its lacunary version. Again, if the choice of the scale function is taken to be a constant different from zero, then we have the notions of ideal lacunary statistical convergence and ideal lacunary summability for function sequences (cf. [24,31,32]).
Let us take α 1 ( m ) = m − λ m + 1 , β 1 ( m ) = m for all m ∈ N and α 2 ( n ) = n − μ n + 1 , β 1 ( n ) = n for all n ∈ N . Then, strong ℐ 2 − α β -relative uniform summability is reduced to strong ℐ 2 − ( λ , μ ) -relative uniform summability. If the scale function is considered to be a constant different from zero, we have the concepts of ideal ( λ , μ ) -statistical convergence and ideal ( λ , μ ) -summability for function sequences (cf. [18,19,33]).

Theorem 1

Suppose that

sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) ≤ K , for a l l i , j ∈ N , ∣ σ ( x , y ) ∣ ≠ 0 .

If a double sequence ( f i j ) of functions on X 2 is ℐ 2 − α β -statistically relatively uniformly convergent to a function f, then it is ℐ 2 − α β -relatively uniformly summable to the function f, but not conversely.

Proof

Let us set

H s t ( ℐ 2 α β ) ≔ ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) ≥ ε

and

H s t ( ℐ 2 α β ) C ≔ ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) < ε ,

with for given ε > 0 and enough large m and n . Also, we obtain

1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∑ i ∈ D m α 1 β 1 ∑ j ∈ D n α 2 β 2 sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) = 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∑ i ∈ D m α 1 β 1 ∑ j ∈ D n α 2 β 2 ( ( i , j ) ∈ H s t ( ℐ 2 α β ) ) sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) + 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∑ i ∈ D m α 1 β 1 ∑ j ∈ D n α 2 β 2 ( ( i , j ) ∈ H s t ( ℐ 2 α β ) C ) sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) ≤ K 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ H s t ( ℐ 2 α β ) ∣ + ε .

From the hypotheses, we have

H s t ( ℐ 2 α β ) 1 ≔ ( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ H s t ( ℐ 2 α β ) ∣ ≥ ε K ∈ ℐ 2 .

If ( m , n ) ∈ N 2 \ H s t ( ℐ 2 α β ) 1 , then

1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∑ i ∈ D m α 1 β 1 ∑ j ∈ D n α 2 β 2 sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) ≤ 2 ε .

Hence, we have

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∑ i ∈ D m α 1 β 1 ∑ j ∈ D n α 2 β 2 sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) ≥ 2 ε ⊂ H s t ( ℐ 2 α β ) 1 ,

and therefore, ( f i j ) is ℐ 2 ‐ α β -relatively uniformly summable to the function f . □

For converse, we consider the following example:

Example 3

Let ℐ 2 = ℐ 2 ϕ , set of all subsets of N 2 with double natural density zero. For each ( i , j ) ∈ N 2 , let α 1 ( m ) ≤ 1 ≤ β 1 ( m ) , α 2 ( n ) ≤ 1 ≤ β 2 ( n ) and define h i j : [ 0 , 1 ] 2 → R by

h i j ( x , y ) = i j 2 x y 2 ; 1 ≤ i ≤ [ β 1 ( m ) − α 1 ( m ) + 1 ] , 1 ≤ j ≤ [ β 2 ( n ) − α 2 ( n ) + 1 ] , ∧ m = 2 l , n = 2 k , l , k ∈ N 3 i j 2 x y 2 2 + i 2 j 3 x 2 y 4 ; otherwise.

One may deduce that s t ( ℐ 2 α β ) – h i j ⇉ h = 0 uniformly on the domain [ 0 , 1 ] 2 , where

σ ( x , y ) = 1 x y 2 , ( x , y ) ∈ ( 0 , 1 ] × ( 0 , 1 ] , 1 , x = 0 or y = 0 .

Indeed, ∃ J = J ( ε ) ,

1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 h i j ( x , y ) σ ( x , y ) ≥ ε = [ β 1 ( m ) − α 1 ( m ) + 1 ] [ β 2 ( n ) − α 2 ( n ) + 1 ] ( β 1 ( m ) − α 1 ( m ) + 1 ) ( β 2 ( n ) − α 2 ( n ) + 1 ) ; m = 2 l , n = 2 k , l , k ∈ N [ J ] ( β 1 ( m ) − α 1 ( m ) + 1 ) ( β 2 ( n ) − α 2 ( n ) + 1 ) ; otherwise. → 0

However,

1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∑ i ∈ D m α 1 β 1 ∑ j ∈ D n α 2 β 2 sup ( x , y ) ∈ X 2 h i j ( x , y ) σ ( x , y ) = [ β 1 ( m ) − α 1 ( m ) + 1 ] ( [ β 1 ( m ) − α 1 ( m ) + 1 ] + 1 ) [ β 2 ( n ) − α 2 ( n ) + 1 ] ( [ β 2 ( n ) − α 2 ( n ) + 1 ] + 1 ) 4 ( β 1 ( m ) − α 1 ( m ) + 1 ) ( β 2 ( n ) − α 2 ( n ) + 1 ) ↛ 0 ,

which means ( h i j ) is not ℐ 2 − α β -relatively uniformly summable to the function h .

4 Ideal relative Korovkin-type approximation

Let the space of all continuous real-valued functions on X 2 be named C ( X 2 ) . As it is known, it is a Banach space with the norm ∥ . ∥ defined by ∥ f ∥ ≔ sup ( x , y ) ∈ X 2 ∣ f ( x , y ) ∣ , f ∈ C ( X 2 ) . In this section, we explore the concept of ℐ 2 - α β -statistical relative uniform convergence to establish a Korovkin-type approximation theorem for a double sequence of positive linear operators acting on the space C ( X 2 ) . Consider a linear operator T : C ( X 2 ) → C ( X 2 ) . For any function f ∈ C ( X 2 ) and point ( x , y ) ∈ X 2 , we write the operator’s value as

T ( f ; x , y ) or equivalently, T ( f ( s , t ) ; x , y ) . Moreover, throughout this article, the test functions are given by

e 0 ( x , y ) = 1 , e 1 ( x , y ) = x , e 2 ( x , y ) = y , and e 3 ( x , y ) = x 2 + y 2 .

Prior to proceeding, we formally review the fundamental Korovkin-type approximation theorems relevant to our study.

Theorem 2

[34] Let ( T i j ) be a double sequence of positive linear operators acting from C ( X 2 ) into itself. Then, for all f ∈ C ( X 2 ) ,

P − lim i j ∥ T i j ( f ) − f ∥ = 0 , on X 2 ,

iff

P − lim i j ∥ T i j ( e z ) − e z ∥ , on X 2 , z = 0 , 1 , 2 , 3 .

Theorem 3

[17] Let ( T i j ) be a double sequence of positive linear operators acting from C ( X 2 ) into itself. Then, for all f ∈ C ( X 2 ) ,

s t 2 α β − lim i j ∥ T i j ( f ) − f ∥ , on X 2

iff

s t 2 α β − lim i j ∥ T i j ( e z ) − e z ∥ , on X 2 , z = 0 , 1 , 2 , 3 .

Now, we give the main approximation result of this section.

Theorem 4

Let ( T i j ) be a double sequence of positive linear operators acting from C ( X 2 ) into itself, and σ and σ z are the scale functions (possibly unbounded). Then, for all f ∈ C ( X 2 ) ,

(3) s t ( ℐ 2 α β ) − T i j ( f ) ⇉ f ( X 2 ; σ ) ,

iff

(4) s t ( ℐ 2 α β ) − T i j ( e z ) ⇉ e z ( X 2 ; σ z ) , z = 0 , 1 , 2 , 3 .

Proof

Based on the assumptions, note that e z ∈ C ( X 2 ) holds for each z = 0 , 1, 2, 3. Hence, condition (3) ensures the validity of condition (4). The main objective now is to establish the converse implication.

As a starting point, since f is continuous on X 2 , it must be bounded over this domain. Thus, there exists a constant K f > 0 such that

∣ f ( x , y ) ∣ ≤ K f ,

where K f = ∥ f ∥ denotes the supremum norm of f on X 2 .

Moreover, the continuity of f implies that for every ε > 0 , there exists a δ > 0 such that

∣ f ( x , y ) − f ( s , t ) ∣ < ε ,

whenever ( x , y ) , ( s , t ) ∈ X 2 and ∣ x − s ∣ < δ , ∣ y − t ∣ < δ .

On the other hand, for all ( x , y ) , ( s , t ) ∈ X 2 satisfying ∣ x − s ∣ > δ and ∣ y − t ∣ > δ , we can estimate:

∣ f ( x , y ) − f ( s , t ) ∣ ≤ 2 K f δ 2 { ( x − s ) 2 + ( y − t ) 2 } .

Consequently, for ( x , y ) , ( s , t ) ∈ X 2 , the following inequality holds:

∣ f ( x , y ) − f ( s , t ) ∣ < ε + 2 K f δ 2 { ( x − s ) 2 + ( y − t ) 2 } .

Given that T i j is both linear and positive, it follows that

∣ T i j ( f ; x , y ) − f ( x , y ) ∣ ≤ T i j ( ∣ f ( x , y ) − f ( s , t ) ∣ ; x , y ) + ∣ f ( x , y ) ∣ ∣ T i j ( e 0 ; x , y ) − e 0 ( x , y ) ∣ ≤ T i j ( ε + 2 K f δ 2 { ( x − s ) 2 + ( y − t ) 2 } ; x , y ) + K f ∣ T i j ( e 0 ; x , y ) − e 0 ( x , y ) ∣

holds for every ( x , y ) ∈ X 2 and i , j ∈ N . Moreover, the following inequality directly follows from the preceding one:

∣ T i j ( f ; x , y ) − f ( x , y ) ∣ ≤ ε + ε + K f + 4 K f δ 2 E 2 ∣ T i j ( e 0 ; x , y ) − e 0 ( x , y ) ∣ + 4 K f δ 2 E ∣ T i j ( e 1 ; x , y ) − e 1 ( x , y ) ∣ + 4 K f δ 2 E ∣ T i j ( e 2 ; x , y ) − e 2 ( x , y ) ∣ + 2 K f δ 2 ∣ T i j ( e 3 ; x , y ) − e 3 ( x , y ) ∣ ,

where E ≔ max { ∣ x ∣ , ∣ y ∣ } . Now, define σ ( x , y ) = max { ∣ σ z ( x , y ) ∣ ; z = 0 , 1 , 2 , 3 } . By multiplying both sides of the preceding inequality by 1 ∣ σ ( x , y ) ∣ , we arrive at the following conclusion:

T i j ( f ; x , y ) − f ( x , y ) σ ( x , y ) ≤ ε ∣ σ ( x , y ) ∣ + K T i j ( e 0 ; x , y ) − e 0 ( x , y ) σ 0 ( x , y ) + T i j ( e 1 ; x , y ) − e 1 ( x , y ) σ 1 ( x , y ) + T i j ( e 2 ; x , y ) − e 2 ( x , y ) σ 2 ( x , y ) + T i j ( e 3 ; x , y ) − e 3 ( x , y ) σ 3 ( x , y ) ,

where K ≔ max ε + K f + 4 K f δ 2 E 2 , 4 K f δ 2 E , 2 K f δ 2 . Thus, taking supremum over ( x , y ) ∈ X 2 , we have

(5) sup ( x , y ) ∈ X 2 T i j ( f ; x , y ) − f ( x , y ) σ ( x , y ) ≤ sup ( x , y ) ∈ X 2 ε ∣ σ ( x , y ) ∣ + K sup ( x , y ) ∈ X 2 T i j ( e 0 ; x , y ) − e 0 ( x , y ) σ 0 ( x , y ) + sup ( x , y ) ∈ X 2 T i j ( e 1 ; x , y ) − e 1 ( x , y ) σ 1 ( x , y )

(6) + sup ( x , y ) ∈ X 2 T i j ( e 2 ; x , y ) − e 2 ( x , y ) σ 2 ( x , y ) + sup ( x , y ) ∈ X 2 T i j ( e 3 ; x , y ) − e 3 ( x , y ) σ 3 ( x , y )

Now, let r > 0 be arbitrary. Select ε > 0 such that sup ( x , y ) ∈ X 2 ε ∣ σ ( x , y ) ∣ < r . Then,

N ≔ ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 T i j ( f ; x , y ) − f ( x , y ) σ ( x , y ) ≥ r

and

N z ≔ ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 T i j ( e z ; x , y ) − e z ( x , y ) σ z ( x , y ) ≥ z − sup ( x , y ) ∈ X 2 ε ∣ σ ( x , y ) ∣ 3 K ,

z = 0 , 1, 2, 3. In light of (5), it is evident that N ⊂ ⋃ 3 z = 0 N z and

1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ N ∣ ≤ 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ N 0 ∣ + 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ N 1 ∣ + 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ N 2 ∣ + 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ N 3 ∣ .

Then, we can write

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ N ∣ ≥ η ⊂ ⋃ z = 0 3 ( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ N z ∣ ≥ η 4 .

By (4),

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ N z ∣ ≥ η 4 ∈ ℐ 2 , for z = 0 , 1 , 2 , 3 .

Hence, by the definition of an ideal,

⋃ z = 0 3 ( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ N z ∣ ≥ η 4 ∈ ℐ 2 , ( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ ∣ N ∣ ≥ η ∈ ℐ 2 .

So, we obtain

s t ( ℐ 2 α β ) − T i j ( f ) ⇉ f ( X 2 ; σ ) .

Hence, the desired conclusion is obtained.□

Moreover, if the scale function is taken to be a nonzero constant, the subsequent result can be directly deduced from Theorem 4.

Corollary 1

Let ( T i j ) be a double sequence of positive linear operators acting from C ( X 2 ) into itself. Then, for all f ∈ C ( X 2 ) ,

s t ( ℐ 2 α β ) − T i j ( f ) ⇉ f , on X 2 ,

if and only if

s t ( ℐ 2 α β ) − T i j ( e z ) ⇉ e , on X 2 , z = 0 , 1 , 2 , 3 .

5 Application of Korovkin-type approximation

Our attention is now paid to an example that shows our main theorem is a non-trivial generalization of the classical and the ideal cases of the Korovkin results.

Example 4

Consider the following Bernstein operators (see [35]) given by

(7) B i j ( f ; x , y ) = ∑ k = 0 i ∑ z = 0 j f k i , l j i k j l x k ( 1 − x ) i − k y l ( 1 − y ) j − l ,

where ( x , y ) ∈ X 2 = [ 0 , 1 ] 2 ; f ∈ C ( X 2 ) . Based on the aforementioned polynomials, we define the following class of positive linear operators on C ( X 2 ) :

(8) T i j ( f ; x , y ) = ( 1 + h i j ( x , y ) ) B i j ( f ; x , y ) .

Here, the auxiliary function h i j ( x , y ) is specified in equation (1). We now examine the behavior of these operators on the test functions:

T i j ( e 0 ; x , y ) = ( 1 + h i j ( x , y ) ) e 0 ( x , y ) , T i j ( e 1 ; x , y ) = ( 1 + h i j ( x , y ) ) e 1 ( x , y ) , T i j ( e 2 ; x , y ) = ( 1 + h i j ( x , y ) ) e 2 ( x , y ) ,

T i j ( e 3 ; x , y ) = ( 1 + h i j ( x , y ) ) e 3 ( x , y ) + x − x 2 i + y − y 2 j .

In the view of s t ( ℐ 2 α β ) − h i j ⇉ h = 0 ( X 2 ; σ ) , σ ( x , y ) is given by (2), we conclude that

s t ( ℐ 2 α β ) − T i j ( e z ) ⇉ e z ( X 2 ; σ ) , for each z = 0 , 1 , 2 , 3 .

Thus, by Theorem 4, we immediately see that

s t ( ℐ 2 α β ) − T i j ( f ) ⇉ f ( X 2 ; σ ) , for all f ∈ C ( X 2 ) .

Unfortunately, since ( h i j ) is neither α β -statistically uniformly convergent nor ℐ 2 − α β -statistically uniformly convergent to the function h = 0 on the interval X 2 , we see that Theorem 3 and Corollary 1 do not work for our operators defined by (8). In addition, one can see that ( h i j ) does not satisfy the classical Korovkin theorem in the Pringsheim’ sense, i.e., Theorem 2 does not work. As a result, it is demonstrated that the presented formulation constitutes a broader generalization than the previously established one.

6 Rate of ℐ 2 - α β -statistical relative uniform convergence

This section aims to analyze the convergence rate corresponding to the ℐ 2 - α β -statistical relative uniform convergence.

Let us recall that the modulus of continuity of a function f ∈ C ( X 2 ) is defined by

ω ( f , δ ) = sup ( x − s ) 2 + ( y − t ) 2 ≤ δ ∣ f ( x , y ) − f ( s , t ) ∣ , ( δ > 0 ) , f ∈ C ( X 2 ) .

Definition 13

Let ( a i j ) denote a positive, non-increasing double sequence. We define that the sequence ( f i j ) converges to f in the sense of ℐ 2 - α β -statistical relative uniform convergence at the rate o ( a i j ) if for any ε > 0 and η > 0 ,

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ × ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) ≥ ε ≥ η a m n ∈ ℐ 2 ,

and written in short as

s t ( ℐ 2 α β ) − ( f i j − f ) = o ( a i j ) ( X 2 ; σ ) .

Definition 14

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ × ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 f i j ( x , y ) − f ( x , y ) σ ( x , y ) ≥ ε a i j ≥ η ∈ ℐ 2 ,

and written in short as

s t ( ℐ 2 α β ) − ( f i j − f ) = o i ( a i j ) ( X 2 ; σ ) .

Lemma 2

Let ( f i j ) and ( h i j ) be double sequences of functions belonging to C ( X 2 ) . Suppose ( α i j ) and ( β i j ) are positive non-increasing double sequences with the property that

s t ( ℐ 2 α β ) − ( f i j − f ) = o ( α i j ) ( X 2 ; σ 0 ) .

and

s t ( ℐ 2 α β ) − ( h i j − h ) = o ( β i j ) ( X 2 ; σ 1 ) ,

where ∣ σ z ( x , y ) ∣ > 0 and σ z ( x , y ) is unbounded, z = 0 , 1 . As a result, the following properties are satisfied:

s t ( ℐ 2 α β ) − ( f i j + h i j ) − ( f + h ) = o ( max { α i j , β i j } ) ( X 2 ; max { ∣ σ z ( x , y ) ∣ ; z = 0 , 1 } ) ,
s t ( ℐ 2 α β ) − ( f i j − f ) ( h i j − h ) = o ( α i j β i j ) ( X 2 ; σ 0 ( x , y ) σ 1 ( x , y ) ) ,
s t ( ℐ 2 α β ) − ( c ( f i j − f ) ) = o ( α i j ) ( X 2 ; σ 0 ( x , y ) ) for any real number c ,
s t ( ℐ 2 α β ) − ∣ f i j − f ∣ = o ( α i j ) ( X 2 ; ∣ σ 0 ( x , y ) ∣ ) .

In addition, a comparable assertion can be established when the notation “ o ” is appropriately replaced by “ o i .”

Proof

The proof is straightforward and so is omitted.□

Lemma 3

Let ( f i j ) and ( h i j ) be sequences of functions in C ( X 2 ) such that 0 ≤ f i j ≤ h i j . Suppose ( α i j ) is a positive non-increasing double sequence with the property that

s t ( ℐ 2 α β ) − h i j = o ( α i j ) ( X 2 ; σ ) .

Then, it follows that

s t ( ℐ 2 α β ) − f i j = o ( α i j ) ( X 2 ; σ ) ,

where ∣ σ ( x , y ) ∣ > 0 and σ ( x , y ) is unbounded. Furthermore, the statement remains valid when the notation “o” is replaced by “ o i .”

Proof

Since the inequalities 0 ≤ f i j ( x , y ) ≤ h i j ( x , y ) are satisfied for every ( x , y ) ∈ X 2 and for all indices ( m , n ) ∈ N 2 , it holds that for arbitrary ε > 0 and η > 0 ,

( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ × ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 f i j ( x , y ) σ ( x , y ) ≥ ε ≥ η α m n ⊂ ( m , n ) ∈ N 2 : 1 ∣ D m α 1 β 1 ∣ ∣ D n α 2 β 2 ∣ × ( i , j ) , i ∈ D m α 1 β 1 and j ∈ D n α 2 β 2 : sup ( x , y ) ∈ X 2 h i j ( x , y ) σ ( x , y ) ≥ ε ≥ η α m n .

Utilizing the relation s t ( ℐ 2 α β ) − h i j = o ( α i j ) on ( X 2 ; σ ) , we arrive at the intended conclusion.□

Consequently, we arrive at the following conclusion.

Theorem 5

Let ( T i j ) denote a double sequence of positive linear operators mapping C ( X 2 ) into itself. In addition, let ( α i j ) and ( β i j ) represent positive and non-increasing double sequences. Suppose that the following assumptions are satisfied:

s t ( ℐ 2 α β ) − ( T i j ( e 0 ) − e 0 ) = o ( α i j ) ( X 2 ; σ 0 ) ,
s t ( ℐ 2 α β ) − ω ( f , δ i j ) = o ( β i j ) ( X 2 ; σ 1 ) , where δ i j ( x , y ) = T i j ( φ ( x , y ) ; x , y ) with φ ( x , y ) ( s , t ) = ( x − s ) 2 + ( y − t ) 2 .

Accordingly, for every f ∈ C ( X 2 ) , we obtain

s t ( ℐ 2 α β ) − ( T i j ( f ) − f ) = o ( γ m n ) ( X 2 ; σ ) ,

where γ m n = max { α i j , β i j , α i j β i j } , and σ ( x , y ) = max { ∣ σ 0 ( x , y ) ∣ , ∣ σ 1 ( x , y ) ∣ , ∣ σ 0 ( x , y ) σ 1 ( x , y ) ∣ } , with ∣ σ z ( x , y ) ∣ > 0 and σ z ( x , y ) is unbounded for z = 0 , 1 .

Proof

Let f ∈ C ( X 2 ) and consider any point ( x , y ) ∈ X 2 . Given that T i j is a positive linear operator and by utilizing the characteristic property of the modulus of continuity, we have

ω ( f , φ ( x , y ) ( s , t ) ) ≤ 1 + φ ( x , y ) ( s , t ) δ 2 ω ( f , δ ) .

It follows that, for any δ ,

∣ T i j ( f ; x , y ) − f ( x , y ) ∣ ≤ T i j ( ∣ f ( x , y ) − f ( s , t ) ∣ ; x , y ) + ∣ f ( x , y ) ∣ ∣ T i j ( e 0 ; x , y ) − e 0 ( x , y ) ∣ ≤ T i j ( ω ( f , φ ( x , y ) ( s , t ) ) ; x , y ) + ∣ f ( x , y ) ∣ ∣ T i j ( e 0 ; x , y ) − e 0 ( x , y ) ∣ ≤ ω ( f , δ ) T i j ( 1 + φ ( x , y ) ( s , t ) δ 2 ; x , y ) + ∣ f ( x , y ) ∣ ∣ T i j ( e 0 ; x , y ) − e 0 ( x , y ) ∣ ≤ ω ( f , δ ) T i j ( e 0 ; x , y ) + 1 δ 2 T i j ( φ ( x , y ) ( s , t ) ; x , y ) + ∣ f ( x , y ) ∣ ∣ T i j ( e 0 ; x , y ) − e 0 ( x , y ) ∣ .

Setting δ ≔ δ i j ( x , y ) = T i j ( φ ( x , y ) ; x , y ) , we may write that

∣ T i j ( f ; x , y ) − f ( x , y ) ∣ ≤ 2 ω ( f , δ i j ) + ω ( f , δ i j ) ∣ T i j ( e 0 ; x , y ) − e 0 ( x , y ) ∣ + M ∣ T i j ( e 0 ; x , y ) − e 0 ( x , y ) ∣ ,

where M = ∥ f ∥ . Taking into account the preceding inequality, we can express the following:

sup ( x , y ) ∈ X 2 T i j ( f ; x , y ) − f ( x , y ) σ ( x , y ) ≤ 2 sup ( x , y ) ∈ X 2 ω ( f , δ i j ) ∣ σ 1 ( x , y ) ∣ + sup ( x , y ) ∈ X 2 ω ( f , δ i j ) ∣ σ 1 ( x , y ) ∣ sup ( x , y ) ∈ X 2 T i j ( e 0 ; x , y ) − e 0 ( x , y ) σ 0 ( x , y ) + M sup ( x , y ) ∈ X 2 T i j ( e 0 ; x , y ) − e 0 ( x , y ) σ 0 ( x , y ) .

Thus, under assumptions ( a ) and ( b ) , and by invoking Lemmas 2 and 3, the result follows.□

The proof of the following theorem is analogous to the proof of Theorem 5, and so is omitted.

Theorem 6

Let ( T i j ) be a double sequence of positive linear operators mapping C ( X 2 ) into itself. Furthermore, let ( α i j ) and ( β i j ) be positive, non-increasing double sequences. Suppose that the following assumptions are satisfied:

s t ( ℐ 2 α β ) − ( T i j ( e 0 ) − e 0 ) = o i ( α i j ) ( X 2 ; σ 0 ) ,
s t ( ℐ 2 α β ) − ω ( f , δ i j ) = o i ( β i j ) ( X 2 ; σ 1 ) , where δ i j ( x , y ) = T i j ( φ ( x , y ) ; x , y ) with φ ( x , y ) ( s , t ) = ( x − s ) 2 + ( y − t ) 2 .

Then, we have, for all f ∈ C ( X 2 ) ,

s t ( ℐ 2 α β ) − ( T i j ( f ) − f ) = o i ( γ m n ) ( X 2 ; σ ) ,

where γ m n = max { α i j , β i j , α i j β i j } and σ ( x , y ) = max { ∣ σ 0 ( x , y ) ∣ , ∣ σ 1 ( x , y ) ∣ , ∣ σ 0 ( x , y ) σ 1 ( x , y ) ∣ } , ∣ σ z ( x , y ) ∣ > 0 and σ z ( x , y ) is unbounded, z = 0 , 1 .

Acknowledgement

The authors express their sincere gratitude to the referees for their careful reading of the manuscript and valuable comments.

Funding information: The authors state no funding involved.
Author contributions: Both authors contributed equally to this work.
Conflict of interest: The authors state no conflict of interest.
Ethical approval: Not applicable.
Data availability statement: Not applicable.

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Received: 2024-12-02

Revised: 2025-05-30

Accepted: 2025-07-03

Published Online: 2025-10-04

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0163

Keywords for this article

Korovkin-type theorem; positive linear operator; ideal relative convergence for double sequences; ℐ2-α β-statistical relative uniform convergence

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