Next Article in Journal
A Non-Standard Finite Difference Scheme for a Diffusive HIV-1 Infection Model with Immune Response and Intracellular Delay
Next Article in Special Issue
Approximation Properties of the Generalized Abel-Poisson Integrals on the Weyl-Nagy Classes
Previous Article in Journal
Optimal Consumption, Investment, and Housing Choice: A Dynamic Programming Approach
Previous Article in Special Issue
Quasi-Density of Sets, Quasi-Statistical Convergence and the Matrix Summability Method
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions

1
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
2
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
3
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan
4
Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy
5
Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, India
*
Author to whom correspondence should be addressed.
Axioms 2022, 11(3), 128; https://doi.org/10.3390/axioms11030128
Submission received: 7 February 2022 / Revised: 2 March 2022 / Accepted: 11 March 2022 / Published: 12 March 2022
(This article belongs to the Special Issue Approximation Theory and Related Applications)

Abstract

:
Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful concepts. We also state and prove two Korovkin-type approximation theorems involving algebraic test functions by using our proposed concepts and methodologies. Furthermore, in order to demonstrate the usefulness of our findings, we consider an illustrative example involving a sequence of positive linear operators in conjunction with the familiar Bernstein polynomials. Finally, in the concluding section, we propose some directions for future research on this topic, which are based upon the core concept of statistical Lebesgue-measurable sequences of functions.

1. Introduction and Motivation

The relatively more familiar theory of ordinary convergence is one of the most important topics of study of sequence spaces. It has indeed gradually progressed to a very high level of development. Two prominent researchers, Fast [1] and Steinhaus [2], independently created a new idea in the theory of sequence spaces, which is known as statistical convergence. This fruitful concept is extremely valuable for studies in various areas of pure and applied mathematical sciences. It is remarkably more powerful than the traditional convergence and has provided a vital area of research in recent years. Furthermore, such a concept is closely related to the study of Real Analysis, Analytic Probability theory and Number theory, and so on. For some recent related developments on this subject, the reader can see, for example, the works in [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18].
Suppose that E N . Moreover, let
E k = { η : η k and η E } .
Then, the natural (or asymptotic) density d ( E ) of E is
d ( E ) = lim k | E k | k = τ ,
where τ is a real and finite number, and | E k | is the cardinality of E k .
A sequence ( u n ) is said to be statistically convergent to α if, for each ϵ > 0 ,
E ϵ = { η : η N and | u η α | ϵ }
has zero natural density (see [1,2]). Thus, for every ϵ > 0 ,
d ( E ϵ ) = lim k | E ϵ | k = 0 .
We write
stat lim k u k = α .
For a closed and bounded interval I : = [ a , b ] R , we define the partition of [ a , b ] as an ordered set that is finite and we denote it as follows:
P : = { ( r 0 , r 1 , , r k ) : a = r 0 < r 1 < < r k = b } .
We now divide the interval [ a , b ] into the following non-overlapping subintervals:
I 1 : = [ r 0 , r 1 ] , I 2 : = [ r 1 , r 2 ] , , I k : = [ r k 1 , r k ] .
The resulting partition P is then given by
P : = { [ r i 1 , r i ] : i = 1 , 2 , 3 , , k } .
Next, in order to find the norm of the partition P , we have
P : = max { r 1 r 0 , r 2 r 1 , r 3 r 2 , , r k r k 1 } .
Let γ i ( i = 1 , 2 , 3 , , k ) be a point that is chosen arbitrarily from each of the subintervals ( I ) i = 1 k . We refer to these points as the tags of the subintervals. We also call the subintervals associated with the tags the tagged partitions of I . We denote it as follows:
P : = { ( [ r i 1 , r i ] ; γ i ) : i = 1 , 2 , 3 , , k } .
Let [ a , b ] R . Suppose that, for each i N , there is a function h i : [ a , b ] R . We thus construct the sequence ( h i ) i N of functions over the closed interval [ a , b ] .
We now define a subsequence ( h i ) i k of functions with respect to the Riemann sum associated with a tagged partition P as follows:
δ ( h i ; P ) : = i = 1 k h ( γ i ) ( r i r i 1 ) .
We next recall the definition of the Riemann integrability.
A sequence ( h k ) k N of functions is Riemann-integrable to h on [ a , b ] if, for each ϵ > 0 , there exists σ ϵ > 0 such that, for any tagged partition P of [ a , b ] with P < σ ϵ , we have
| δ ( h k ; P ) h | < ϵ .
The definition of statistically Riemann-integrable functions is given as follows.
Definition 1.
A sequence ( h k ) k N of functions is statistically Riemann-integrable to h on [ a , b ] if, for every ϵ > 0 and for each x [ a , b ] , there exists σ ϵ > 0 , and for any tagged partition P of [ a , b ] with P < σ ϵ , the set
E ϵ = { η : η N and | δ ( h η ; P ) h | ϵ }
has zero natural density. That is, for every ϵ > 0 ,
d ( E ϵ ) = lim k | E ϵ | k = 0 .
We write
stat Rie lim k δ ( h k ; P ) = h .
By making use of Definition 1, we first establish an inclusion theorem as Theorem 1 below.
Theorem 1.
If a sequence of functions ( h k ) is Riemann-integrable to h over [ a , b ] , then ( h k ) is statistically Riemann-integrable to the same function h over [ a , b ] .
Proof. 
Given ϵ > 0 , there exists σ ϵ > 0 . Suppose that P is any tagged partition of [ a , b ] such that P < σ ϵ . Then
| δ ( h k ; P ) h | < ϵ .
Since, for each ϵ > 0 , P is any tagged partition of [ a , b ] such that P < σ ϵ , so we have
lim k 1 k | { η : η N and | δ ( h k ; P ) h | ϵ } | lim k | δ ( h k ; P ) h | < ϵ .
Consequently, by Definition 1, we get
stat Rie lim k δ ( h k ; P ) = h ,
which completes the proof of Theorem 1. □
Remark 1.
In order to demonstrate that the converse of Theorem 1 is not true, we consider Example 1 below.
Example 1.
Let h k : [ 0 , 1 ] R be a sequence of functions defined by
h k ( x ) = 1 2 ( x Q [ 0 , 1 ] ; k = j 2 , j N ) 1 n ( otherwise ) .
It is easily seen that the sequence ( h k ) of functions is statistically Riemann-integrable to 0 over the closed interval [ 0 , 1 ] , but it is not Riemann-integrable (in the usual sense) over [ 0 , 1 ] .
Motivated mainly by the above-mentioned investigations and developments, we introduce and study the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability of sequences of real-valued functions. We first prove an inclusion theorem connecting these two potentially useful concepts. We then state and prove two Korovkin-type approximation theorems with algebraic test functions based on the methodologies and techniques that we have adopted here. Furthermore, we consider an illustrative example involving a positive linear operator in conjunction with the familiar Bernstein polynomials, which shows the effectiveness of our findings. Finally, based upon the core concept of statistical Lebesgue-measurable sequences of functions, we suggest some possible directions for future research on this topic in the concluding section of our study.

2. Deferred Weighted Statistical Riemann Integrability

Let ( ϕ k ) and ( φ k ) be sequences of non-negative integers with the regularity conditions given
ϕ k < φ k and lim k φ k = + .
Moreover, let ( p i ) be a sequence of non-negative real numbers with
P k = i = ϕ k + 1 φ k p i .
We then define the deferred weighted summability mean for δ ( h k ; P ) associated with tagged partition P as follows:
W ( δ ( h k ; P ) ) = 1 P k ϱ = ϕ k + 1 φ k p ϱ δ ( h ϱ ; P ) .
We now present the following definitions for our proposed study.
Definition 2.
A sequence ( h k ) k N of functions is said to be deferred weighted statistically Riemann-integrable to h on [ a , b ] if, for all ϵ > 0 , there exists σ ϵ > 0 , and for any tagged partition P of [ a , b ] with P < σ ϵ , the following set
{ η : η P k and p η | δ ( h η ; P ) h | ϵ }
has zero natural density. Thus, for every ϵ > 0 , we have
lim k | { η : η P k and p η | δ ( h η ; P ) k | ϵ } | P k = 0 .
We write
DWR stat lim k δ ( h k ; P ) = h .
Definition 3.
A sequence ( h k ) k N of functions is said to statistically deferred weighted Riemann summable to h on [ a , b ] if, for all ϵ > 0     σ ϵ > 0 and for any tagged partition P of [ a , b ] with P < σ ϵ , the set
{ η : η k and | W ( δ ( h η ; P ) ) h | ϵ }
has zero natural density. Thus, for all ϵ > 0 , we have
lim k | { η : η k and | W ( δ ( h η ; P ) ) h | ϵ } | k = 0 .
We write
stat DWR lim k δ ( h k ; P ) = h .
An inclusion theorem between the two new potentially useful notions in Definitions 2 and 3 is now given by Theorem 2 below.
Theorem 2.
If the sequence ( h k ) k N of functions is deferred weighted statistically Riemann-integrable to a function h over [ a , b ] , then it is statistically deferred weighted Riemann summable to the same function h over [ a , b ] , but not conversely.
Proof. 
Suppose that the sequence ( h k ) k N is deferred weighted statistically Riemann-integrable to a function h on [ a , b ] . Then, by Definition 2, we have
lim k | { η : η P k and p η | δ ( h η ; P ) h | ϵ } | P k = 0 .
Now, if we choose the two sets as follows,
O ϵ = { η : η P k and p η | δ ( h η ; P ) h | ϵ }
and
O ϵ c = { η : η P k and p η | δ ( h η ; P ) h | < ϵ } ,
then we have
W ( δ ( h k ; P ) ) h = 1 P k ϱ = ϕ k + 1 φ k p ϱ δ ( h ϱ ; P ) h 1 P k ϱ = ϕ k + 1 φ k p ϱ δ ( h ϱ ; P ) h + 1 P k ϱ = ϕ k + 1 φ k p ϱ h h 1 P k ϱ = ϕ k + 1 ( η O ϵ ) φ k p ϱ δ ( h ϱ ; P ) h + 1 P k ϱ = ϕ k + 1 ( η O ϵ c ) φ k p ϱ δ ( h ϱ ; P ) h + | h | 1 P k ϱ = ϕ k + 1 φ k p ϱ 1 1 P k O ϵ + 1 P k | O ϵ c | .
We thus obtain
| W ( δ ( h k ; P ) ) h | < ϵ .
Hence, clearly, the sequence of functions ( h k ) is statistically deferred weighted Riemann-summable to h over [ a , b ] . □
The following example shows that the converse statement of Theorem 2 is not true.
Example 2.
Let h k : [ 0 , 1 ] R be a sequence of functions of the form given by
h k ( x ) = 0 ( x Q [ 0 , 1 ] ; k is even ) 1 ( x R Q [ 0 , 1 ] ; k is odd ) ,
where
ϕ k = 2 k φ k = 4 k and p k = 1 .
The above-specified sequence ( h k ) of functions trivially indicates that it is neither Riemann-integrable nor deferred weighted statistically Riemann-integrable. However, as per our proposed mean (2), it is easy to see that
W ( δ ( h k ; P ) ) = 1 φ k ϕ k ϱ = ϕ k + 1 φ k δ ( h ϱ ; P ) = 1 2 k m = 2 k + 1 4 k δ ( h ϱ ; P ) = 1 2 .
Thus, clearly, the sequence ( h k ) of functions has deferred weighted Riemann sum 1 2 under the tagged partition P . Therefore, the sequence ( h k ) of functions is statistically deferred weighted Riemann-summable to 1 2 over [ 0 , 1 ] , but it is not deferred weighted statistically Riemann-integrable over [ 0 , 1 ] .

3. Korovkin-Type Approximation Theorems via the W ( δ ( h k ; P ) ) -Mean

Many researchers have worked toward extending (or generalizing) the approximation-theoretic aspects of the Korovkin-type approximation theorems in several different areas of mathematics, such as (for example) probability space, measurable space, sequence spaces, and so on. In Real Analysis, Harmonic Analysis and other related fields, this notion is immensely useful. In this regard, we have chosen to refer the interested reader to the recent works (see, for example, [19,20,21,22,23,24,25,26,27,28]).
Let C [ 0 , 1 ] be the space of all continuous real-valued functions defined on [ 0 , 1 ] . Suppose also that it is a Banach space with the norm . . Then, for h C [ 0 , 1 ] , the norm of h is given by
h = sup { | h ( ρ ) | : 0 ρ 1 } .
We say that G j : C [ 0 , 1 ] C [ 0 , 1 ] is a sequence of positive linear operators, if
G j ( h ; ρ ) 0 as h 0 .
Now, in view of our above-proposed definitions, we state and prove the following Korovkin-type approximation theorems.
Theorem 3.
Let G j : C [ 0 , 1 ] C [ 0 , 1 ] be a sequence of positive linear operators. Then, for h C [ 0 , 1 ] ,
DWR stat lim j G j ( h ; ρ ) h ( ρ ) = 0
if and only if
DWR stat lim j G j ( 1 ; ρ ) 1 = 0 ,
DWR stat lim j G j ( ρ ; ρ ) ρ = 0
and
DWR stat lim j G j ( ρ 2 ; ρ ) ρ 2 = 0 .
Proof. 
Since each of the following functions
h 0 ( ρ ) = 1 , h 1 ( ρ ) = 2 ρ and h 2 ( ρ ) = 3 ρ 2
belongs to C [ 0 , 1 ] and is continuous on [ 0 , 1 ] , the implication given by (4) obviously implies (5) to (7).
In order to complete the proof of Theorem 3, we first assume that the conditions (5) to (7) hold true. If h C [ 0 , 1 ] , then there exists a constant L > 0 such that
| h ( ρ ) | L ( ρ [ 0 , 1 ] ) .
We thus find that
| h ( r ) h ( ρ ) | 2 L ( r , ρ [ 0 , 1 ] ) .
Clearly, for given ϵ > 0 , there exists δ > 0 such that
| f ( r ) f ( ρ ) | < ϵ
whenever
| r ρ | < δ for all r , ρ [ 0 , 1 ] .
If we now choose
μ 1 = μ 1 ( r , ρ ) = ( 2 r 2 ρ ) 2 .
If
| r ρ | δ ,
then we obtain
| h ( r ) h ( ρ ) | < 2 L θ 2 μ 1 ( r , ρ ) .
Thus, from Equations (9) and (10), we get
| h ( r ) h ( ρ ) | < ϵ + 2 L θ 2 μ 1 ( r , ρ ) ,
which implies that
ϵ 2 L θ 2 μ 1 ( r , ρ ) h ( r ) h ( ρ ) ϵ + 2 L θ 2 μ 1 ( r , ρ ) .
Now, since G m ( 1 ; ρ ) is monotone and linear, by applying the operator G m ( 1 ; ρ ) to the inequality (11), we get
G m ( 1 ; ρ ) ϵ 2 L θ 2 μ 1 ( r , ρ ) G m ( 1 ; ρ ) h ( r ) h ( ρ ) G m ( 1 ; ρ ) ϵ + 2 L θ 2 μ 1 ( r , ρ ) .
We note that ρ is fixed, and so h ( ρ ) is a constant number. Therefore, we have
ϵ G m ( 1 ; ρ ) 2 L θ 2 G m ( μ 1 ; ρ ) G m ( h ; ρ ) h ( ρ ) G m ( 1 ; ρ ) ϵ G m ( 1 ; ρ ) + 2 L θ 2 G m ( μ 1 ; ρ ) .
We also know that
G m ( h ; ρ ) h ( ρ ) = [ G m ( h ; ρ ) h ( ρ ) G m ( 1 ; ρ ) ] + h ( ρ ) [ G m ( 1 ; ρ ) 1 ] .
Thus, by using (12) and (13), we obtain
G m ( h ; ρ ) h ( ρ ) < ϵ G m ( 1 ; ρ ) + 2 L θ 2 G m ( μ 1 ; ρ ) + h ( ρ ) [ G m ( 1 ; ρ ) 1 ] .
We now estimate G m ( μ 1 ; ρ ) as follows:
G m ( μ 1 ; ρ ) = G m ( ( 2 r 2 ρ ) 2 ; ρ ) = G m ( 2 r 2 8 ρ r + 4 ρ 2 ; ρ ) = G m ( 4 r 2 ; ρ ) 8 t G m ( r ; ρ ) + 4 ρ 2 G m ( 1 ; ρ ) = 4 [ G m ( r 2 ; ρ ) ρ 2 ] 8 t [ G m ( r ; ρ ) ρ ] + 4 ρ 2 [ G m ( 1 ; ρ ) 1 ] ,
so that, in view of (14), we obtain
G m ( h ; ρ ) h ( ρ ) < ϵ G m ( 1 ; ρ ) + 2 L θ 2 { 4 [ G m ( r 2 ; ρ ) ρ 2 ] 8 ρ [ G m ( r ; ρ ) ρ ] + 4 ρ 2 [ G m ( 1 ; ρ ) 1 ] } + h ( ρ ) [ G m ( 1 ; ρ ) 1 ] . = ϵ [ G m ( 1 ; ρ ) 1 ] + ϵ + 2 L θ 2 { 4 [ G m ( r 2 ; ρ ) ρ 2 ] 8 ρ [ G m ( r ; ρ ) ρ ] + 4 ρ 2 [ G m ( 1 ; ρ ) 1 ] } + h ( ρ ) [ G m ( 1 ; ρ ) 1 ] .
Furthermore, since ϵ > 0 is arbitrary, we can write
| G m ( h ; ρ ) h ( ρ ) | ϵ + ϵ + 8 L θ 2 + L | G m ( 1 ; ρ ) 1 | + 16 L θ 2 | G m ( r ; ρ ) ρ | + 8 L θ 2 | G m ( r 2 ; ρ ) ρ 2 | A ( | G m ( 1 ; ρ ) 1 | + | G m ( r ; ρ ) ρ | + | G m ( r 2 ; ρ ) ρ 2 | ) ,
where
A = max ϵ + 8 L θ 2 + L , 16 L θ 2 , 8 L θ 2 .
Now, for a given ω > 0 , there exists ϵ > 0   ( ϵ < ω ) such that
T m ( ρ ; ω ) = m : m P k and p m G m ( h ; ρ ) h ( ρ ) ω .
Furthermore, for ν = 0 , 1 , 2 , we have
T ν , m ( ρ ; ω ) = m : m P k and p m G m ( h ; ρ ) h ν ( ρ ) ω ϵ 3 A ,
so that
T m ( ρ ; ω ) ν = 0 2 T ν , m ( ρ ; ω ) .
Clearly, we obtain
T m ( ρ ; ω ) C [ 0 , 1 ] P k ν = 0 2 T ν , m ( ρ ; ω ) C [ 0 , 1 ] P k .
Now, using the above assumption about the implications in (5) to (7) and by Definition 2, the right-hand side of (16) tends to zero as n . Consequently, we get
lim k T m ( ρ ; ω ) C [ 0 , 1 ] P k = 0 ( δ , ω > 0 ) .
Therefore, the implication (4) holds true. □
Theorem 4.
Let G j : C [ 0 , 1 ] C [ 0 , 1 ] be a sequence of positive linear operators. Then, for h C [ 0 , 1 ] ,
stat DWR lim j G j ( h ; ρ ) h ( ρ ) = 0
if and only if
stat DWR lim j G j ( 1 ; ρ ) 1 = 0 ,
stat DWR lim j G j ( ρ ; ρ ) ρ = 0
and
stat DWR lim j G j ( ρ 2 ; ρ ) ρ 2 = 0 .
Proof. 
The proof of Theorem 4 is similar to the proof of Theorem 3. Therefore, we choose to skip the details involved. □
In view of Theorem 4, here, we consider an illustrative example. In this connection, we now recall the following operator:
ρ ( 1 + ρ D ) D = d d ρ ,
which was used by Al-Salam [29] and, more recently, by Viskov and Srivastava [30].
Example 3.
Consider the Bernstein polynomials B n ( h ; β ) on C [ 0 , 1 ] given by
B k ( h ; β ) = ϱ = 0 k f ϱ k k ϱ β ϱ ( 1 β ) k ϱ ( β [ 0 , 1 ] ; k = 0 , 1 , · · · ) .
Here, in this example, we introduce the positive linear operators on C [ 0 , 1 ] under the composition of the Bernstein polynomials and the operators given by (21) as follows:
G ϱ ( h ; β ) = [ 1 + h ϱ ] β ( 1 + β D ) B ϱ ( h ; β ) ( h C [ 0 , 1 ] ) ,
where ( h ϱ ) is the same as mentioned in Example 2.
We now estimate the values of each of the testing functions 1, β and β 2 by using our proposed operators (23) as follows:
G ϱ ( 1 ; β ) = [ 1 + h ϱ ] β ( 1 + β D ) 1 = [ 1 + h ϱ ] β ,
G ϱ ( t ; β ) = [ 1 + h ϱ ] β ( 1 + β D ) β = [ 1 + h ϱ ] β ( 1 + β )
and
G ϱ ( t 2 ; β ) = [ 1 + h ϱ ] β ( 1 + β D ) β 2 + β ( 1 β ) ϱ = [ 1 + h ϱ ] β 2 2 3 β ϱ .
Consequently, we have
stat DWR lim ϱ G ϱ ( 1 ; β ) 1 = 0 ,
stat DWR lim ϱ G ϱ ( β ; β ) β = 0
and
stat DWR lim ϱ G ϱ ( β 2 ; β ) β 2 = 0 ,
that is, the sequence G ϱ ( h ; β ) satisfies the conditions (18) to (20). Therefore, by Theorem 4, we have
stat DWR lim ϱ G ϱ ( h ; β ) h = 0 .
Hence, the given sequence ( h k ) of functions mentioned in Example 2 is statistically deferred weighted Riemann-summable, but not deferred weighted statistically Riemann-integrable. Therefore, our above-proposed operators defined by (23) satisfy Theorem 4. However, they do not satisfy for statistical versions of deferred weighted Riemann-integrable sequence of functions (see Theorem 3).

4. Concluding Remarks and Directions for Further Research

In this concluding section of our present investigation, we further observe the potential usefulness of our Theorem 4 over Theorem 3 as well as over the classical versions of the Korovkin-type approximation theorems.
Remark 2.
Let us consider the sequence ( h ϱ ) ϱ N of functions in Example 2. Suppose also that ( h ϱ ) is statistically deferred weighted Riemann-summable, so that
stat DWR lim ϱ δ ( h ϱ ; P ) = 1 2 on [ 0 , 1 ] .
We then find that
stat DWR lim k G k ( h ν ; ρ ) f ν ( ρ ) = 0 ( ν = 0 , 1 , 2 ) .
Thus, by Theorem 4, we immediately get
stat DWR lim j G k ( h ; ρ ) h ( ρ ) = 0 ,
where
h 0 ( ρ ) = 1 , h 1 ( ρ ) = ρ and h 2 ( ρ ) = ρ 2 .
Now, the given sequence ( h k ) of functions is statistically deferred weighted Riemann-summable, but neither deferred weighted statistically Riemann-integrable nor classically Riemann-integrable. Therefore, our Korovkin-type approximation Theorem 4 properly works under the operators defined in the Equation (23), but the classical as well as statistical versions of the deferred weighted Riemann-integrable sequence of functions do not work for the same operators. Clearly, this observation leads us to the fact that our Theorem 4 is a non-trivial extension of Theorem 3 as well as the classical Korovkin-type approximation theorem [31].
Remark 3.
Motivated by some recently published results by Jena et al. [32] and Srivastava et al. [33], we choose to draw the attention of the interested readers toward the potential for further research associated with the analogous notion of statistical Lebesgue-measurable sequences of functions.

Author Contributions

Formal analysis, H.M.S. and S.K.P.; Investigation, B.B.J.; Methodology, S.K.P.; Supervision, H.M.S. and S.K.P.; Writing-original draft, B.B.J.; Writing—review & editing, H.M.S. and S.K.P. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

  1. Fast, H. Sur la convergence statistique. Colloq. Math. 1951, 2, 241–244. [Google Scholar] [CrossRef]
  2. Steinhaus, H. Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 1951, 2, 73–74. [Google Scholar]
  3. Akdag, S. Weighted equi-statistical convergence of the Korovkin-type approximation theorems. Results Math. 2017, 72, 1073–1085. [Google Scholar] [CrossRef]
  4. Balcerzak, M.; Dems, K.; Komisarski, A. Statistical convergence and ideal convergence for sequences of functions. J. Math. Anal. Appl. 2007, 328, 715–729. [Google Scholar] [CrossRef] [Green Version]
  5. Braha, N.L.; Loku, V.; Srivastava, H.M. Λ2-Weighted statistical convergence and Korovkin and Voronovskaya type theorems. Appl. Math. Comput. 2015, 266, 675–686. [Google Scholar] [CrossRef]
  6. Et, M.; Baliarsingh, P.; Küçxuxkaslan, H.S.K.M. On μ-deferred statistical convergence and strongly deferred summable functions. Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat. (RACSAM) 2021, 115, 1–14. [Google Scholar] [CrossRef]
  7. Ghosal, S.; Mandal, S. Rough weighted I-αβ-statistical convergence in locally solid Riesz spaces. J. Math. Anal. Appl. 2021, 506, 125681. [Google Scholar] [CrossRef]
  8. Ghosal, S.; Banerjee, M. Rough weighted statistical convergence on locally solid Riesz spaces. Positivity 2021, 25, 1789–1804. [Google Scholar] [CrossRef]
  9. Guessab, A.; Schmeisser, G. Convexity results and sharp error estimates in approximate multivariate integration. Math. Comput. 2003, 73, 1365–1384. [Google Scholar] [CrossRef]
  10. Alotaibi, A. Generalized weighted statistical convergence for double sequences of fuzzy numbers and associated Korovkin-type approximation theorem. J. Funct. Spaces 2020, 2020, 9298650. [Google Scholar] [CrossRef]
  11. Özger, F. Applications of generalized weighted statistical convergence to approximation theorems for functions of one and two variables. Numer. Funct. Anal. Optim. 2020, 41, 1990–2006. [Google Scholar] [CrossRef]
  12. Demirci, K.; Dirik, F.; Yıldız, S. Deferred Nörlund statistical relative uniform convergence and Korovkin-type approximation theorem. Commun. Fac. Sci. Univ. Ank. Ser. A Math. Statist. 2021, 70, 279–289. [Google Scholar] [CrossRef]
  13. Mohiuddine, S.A.; Hazarika, B. On strongly almost generalized difference lacunary ideal convergent sequences of fuzzy numbers. J. Comput. Anal. Appl. 2017, 23, 925–936. [Google Scholar]
  14. Móricz, F. Tauberian conditions under which statistical convergence follows from statistical summability (C,1). J. Math. Anal. Appl. 2002, 275, 277–287. [Google Scholar] [CrossRef] [Green Version]
  15. Agrawal, P.N.; Acu, A.-M.; Chauhan, R.; Garg, T. Approximation of Bögel continuous functions and deferred weighted A-statistical convergence by Bernstein-Kantorovich type operators on a triangle. J. Math. Inequal. 2021, 15, 1695–1711. [Google Scholar] [CrossRef]
  16. Saini, K.; Raj, K.; Mursaleen, M. Deferred Cesàro and deferred Euler equi-statistical convergence and its applications to Korovkin-type approximation theorem. Internat. J. Gen. Syst. 2021, 50, 567–579. [Google Scholar] [CrossRef]
  17. Söylemez, D. A Korovkin type approximation theorem for Balázs type Bleimann, Butzer and Hahn operators via power series statistical convergence. Math. Slovaca 2022, 72, 153–164. [Google Scholar] [CrossRef]
  18. Turan, C.; Duman, O. Fundamental properties of statistical convergence and lacunary statistical convergence on time scales. Filomat 2017, 31, 4455–4467. [Google Scholar] [CrossRef]
  19. Altomare, F. Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 2010, 5, 92–164. [Google Scholar]
  20. Braha, N.L. Some weighted equi-statistical convergence and Korovkin-type theorem. Results Math. 2016, 70, 433–446. [Google Scholar] [CrossRef]
  21. Braha, N.L.; Srivastava, H.M.; Et, M. Some weighted statistical convergence and associated Korovkin and Voronovskaya type theorems. J. Appl. Math. Comput. 2021, 65, 429–450. [Google Scholar] [CrossRef]
  22. Guessab, A.; Schmeisser, G. Two Korovkin-type theorems in multivariate approximation. Banach J. Math. Anal. 2008, 2, 121–128. [Google Scholar] [CrossRef]
  23. Jena, B.B.; Paikray, S.K.; Dutta, H. On various new concepts of statistical convergence for sequences of random variables via deferred Cesàro mean. J. Math. Anal. Appl. 2020, 487, 123950. [Google Scholar] [CrossRef]
  24. Karakuş, S.; Demirci, K.; Duman, O. Equi-statistical convergence of positive linear operators. J. Math. Anal. Appl. 2008, 339, 1065–1072. [Google Scholar] [CrossRef] [Green Version]
  25. Çınar, S.; Yıldız, S.; Demirci, K. Korovkin type approximation via triangular A-statistical convergence on an infinite interval. Turk. J. Math. 2021, 45, 929–942. [Google Scholar] [CrossRef]
  26. Parida, P.; Paikray, S.K.; Jena, B.B. Generalized deferred Cesàro equi-statistical convergence and analogous approximation theorems. Proyecciones J. Math. 2020, 39, 307–331. [Google Scholar] [CrossRef]
  27. Srivastava, H.M.; Jena, B.B.; Paikray, S.K. Statistical probability convergence via the deferred Nörlund mean and its applications to approximation theorems. Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat. (RACSAM) 2020, 114, 1–14. [Google Scholar] [CrossRef]
  28. Srivastava, H.M.; Jena, B.B.; Paikray, S.K. Statistical deferred Nörlund summability and Korovkin-type approximation theorem. Mathematics 2020, 8, 636. [Google Scholar] [CrossRef] [Green Version]
  29. Al-Salam, W.A. Operational representations for the Laguerre and other polynomials. Duke Math. J. 1964, 31, 127–142. [Google Scholar] [CrossRef]
  30. Viskov, O.V.; Srivastava, H.M. New approaches to certain identities involving differential operators. J. Math. Anal. Appl. 1994, 186, 1–10. [Google Scholar] [CrossRef] [Green Version]
  31. Korovkin, P.P. Convergence of linear positive operators in the spaces of continuous functions. Dokl. Akad. Nauk. SSSR (New Ser.) 1953, 90, 961–964. (In Russian) [Google Scholar]
  32. Jena, B.B.; Paikray, S.K.; Dutta, H. A new approach to Korovkin-type approximation via deferred Cesàro statistical measurable convergence. Chaos Solitons Fractals 2021, 148, 111016. [Google Scholar] [CrossRef]
  33. Srivastava, H.M.; Jena, B.B.; Paikray, S.K. Statistical Riemann and Lebesgue integrable sequence of functions with Korovkin-type approximation theorems. Axioms 2021, 10, 229. [Google Scholar] [CrossRef]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Share and Cite

MDPI and ACS Style

Srivastava, H.M.; Jena, B.B.; Paikray, S.K. Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions. Axioms 2022, 11, 128. https://doi.org/10.3390/axioms11030128

AMA Style

Srivastava HM, Jena BB, Paikray SK. Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions. Axioms. 2022; 11(3):128. https://doi.org/10.3390/axioms11030128

Chicago/Turabian Style

Srivastava, Hari Mohan, Bidu Bhusan Jena, and Susanta Kumar Paikray. 2022. "Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions" Axioms 11, no. 3: 128. https://doi.org/10.3390/axioms11030128

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop