Fibonacci statistical convergence and Korovkin type approximation theorems
- Murat Kirişci^{1}Email authorView ORCID ID profile and
- Ali Karaisa^{2}
https://doi.org/10.1186/s13660-017-1503-z
© The Author(s) 2017
Received: 12 July 2017
Accepted: 6 September 2017
Published: 19 September 2017
Abstract
The purpose of this paper is twofold. First, the definition of new statistical convergence with Fibonacci sequence is given and some fundamental properties of statistical convergence are examined. Second, we provide various approximation results concerning the classical Korovkin theorem via Fibonacci type statistical convergence.
Keywords
MSC
1 Introduction
1.1 Densities and statistical convergence
Let A be a subset of positive integers. We consider the interval \([1,n]\) and select an integer in this interval, randomly. Then the ratio of the number of elements of A in \([1,n]\) to the total number of elements in \([1,n]\) belongs to A, probably. For \(n\rightarrow \infty\), if this probability exists, that is, this probability tends to some limit, then this limit is used as the asymptotic density of the set A. Let us mention that the asymptotic density is a kind of probability of choosing a number from the set A.
Now, we give some definitions and properties of asymptotic density.
The set of positive integers will be denoted by \(\mathbb{Z^{+}}\). Let A and B be subsets of \(\mathbb{Z}^{+}\). If the symmetric difference \(A\Delta B\) is finite, then we can say A is asymptotically equal to B and denote \(A\sim B\). Freedman and Sember introduced the concept of a lower asymptotic density and defined the concept of convergence in density, in [1].
Definition 1.1
[1]
- i.
\(f(A)=f(B)\) if \(A\sim B\);
- ii.
\(f(A)+f(B)\leq f(A\cup B)\) if \(A\cap B=\emptyset\);
- iii.
\(f(A)+f(B)\leq1+ f(A\cap B)\) for all A;
- iv.
\(f(\mathbb{Z^{+}})=1\).
We can define the upper density based on the definition of lower density as follows.
Let f be any density. Then, for any set of natural numbers A, the function f̅ is said to be upper density associated with f if \(\overline{f}(A)=1-f(\mathbb{Z}^{+} \backslash A)\).
The study of statistical convergence was initiated by Fast [2]. Schoenberg [3] studied statistical convergence as a summability method and listed some of the elementary properties of statistical convergence. Both of these mathematicians mentioned that if a bounded sequence is statistically convergent to L, then it is Cesàro summable to L. Statistical convergence also arises as an example of ‘convergence in density’ as introduced by Buck [4]. In [5], Zygmund called this concept ‘almost convergence’ and established the relation between statistical convergence and strong summability. The idea of statistical convergence has been studied in different branches of mathematics such as number theory [6], trigonometric series [5], summability theory [1], measure theory [7] and Hausdorff locally convex topological vector spaces [8]. The concept of αβ-statistical convergence was introduced and studied by Aktuǧlu [9]. In [10], Karakaya and Karaisa extended the concept of αβ-statistical convergence. Also, they introduced the concept of weighted αβ-statistical convergence of order γ, weighted αβ-summability of order γ and strongly weighted αβ-summable sequences of order γ in [10]. In [11], Braha gave a new weighted equi-statistical convergence and proved the Korovkin type theorems using the new definition.
Definition 1.2
A real numbers sequence \(x=(x_{k})\) is statistically convergent to L provided that for every \(\varepsilon>0\) the set \(\{n\in \mathbb{N}: \vert x_{n}-L\vert \geq\varepsilon\}\) has natural density zero. The set of all statistically convergent sequences is denoted by S. In this case, we write \(S-\lim x=L\) or \(x_{k}\rightarrow L(S)\).
Definition 1.3
[12]
It can be seen from the definition that statistical convergence is a generalization of the usual notion of convergence that parallels the usual theory of convergence.
Fridy [12] introduced a new notation for facilitation: If \(x=(x_{n})\) is a sequence that satisfies some property P for all n except a set of natural density zero, then we say that \(x=(x_{n})\) satisfies P for ‘almost all n’, and we abbreviate ‘a.a.n’. In [12], Fridy proved the following theorem.
Theorem 1.4
- i.
x is a statistically convergent sequence;
- ii.
x is a statistically Cauchy sequence;
- iii.
x is a sequence for which there is a convergent sequence y such that \(x_{n}=y_{n}\) for a.a.n.
1.2 Fibonacci numbers and Fibonacci matrix
Definition 1.5
From this definition, it means that the first two numbers in Fibonacci sequence are either 1 and 1 (or 0 and 1) depending on the chosen starting point of the sequence and all subsequent numbers is the sum of the previous two. That is, we can choose \(f_{1}=f_{2}=1\) or \(f_{0}=0\), \(f_{1}=1\).
Fibonacci sequence was initiated in the book Liber Abaci of Fibonacci which was written in 1202. However, the sequence is based on older history. The sequence had been described earlier as Virahanka numbers in Indian mathematics [14]. In Liber Abaci, the sequence starts with 1, nowadays the sequence begins either with \(f_{0}=0\) or with \({f_{1}=1}\).
1.3 Approximation theory
Korovkin type approximation theorems are practical tools to check whether a given sequence \((A_{n})_{n\geq1}\) of positive linear operators on \(C[a,b]\) of all continuous functions on the real interval \([a,b]\) is an approximation process. That is, these theorems present a variety of test functions which provide that the approximation property holds on the whole space if it holds for them. Such a property was determined by Korovkin [16] in 1953 for the functions 1, x and \(x^{2}\) in the space \(C[a,b]\) as well as for the functions 1, cos and sin in the space of all continuous 2π-periodic functions on the real line.
Until the study of Gadjiev and Orhan [17], there was no study related to statistical convergence and approximation theory. In [17], Korovkin type approximation theorems were proved by using the idea of statistical convergence. Some of the examples of approximation theory and statistical convergence studies can be seen in [9, 10, 18–24].
2 Methods
In the theory of numbers, there are many different definitions of density. It is well known that the most popular of these definitions is asymptotic density. However, asymptotic density does not exist for all sequences. New densities have been defined to fill those gaps and to serve different purposes.
The asymptotic density is one of the possibilities to measure how large a subset of the set of natural numbers is. We know intuitively that positive integers are much more than perfect squares. Because every perfect square is positive and many other positive integers exist besides. However, the set of positive integers is not in fact larger than the set of perfect squares: both sets are infinite and countable and can therefore be put in one-to-one correspondence. Nevertheless, if one goes through the natural numbers, the squares become increasingly scarce. It is precisely in this case that natural density helps us and makes this intuition precise.
The Fibonacci sequence was firstly used in the theory of sequence spaces by Kara and Başarır [25]. Afterward, Kara [15] defined the Fibonacci difference matrix F̂ by using the Fibonacci sequence \((f_{n})\) for \(n\in\{0,1,\ldots\}\) and introduced the new sequence spaces related to the matrix domain of F̂.
Following [25] and [15], high quality papers have been produced on the Fibonacci matrix by many mathematicians [26–36].
In this paper, by combining the definitions of Fibonacci sequence and statistical convergence, we obtain a new concept of statistical convergence, which will be called Fibonacci type statistical convergence. We examine some basic properties of new statistical convergence defined by Fibonacci sequences. Henceforth, we get an analogue of the classical Korovkin theorem by using the concept of Fibonacci type statistical convergence.
It will be shown that if X is a Banach space, then for a closed subset of X, which is denoted by A, Fibonacci type space A is closed in Fibonacci type space X. We will give the definitions of Fibonacci statistically Cauchy sequence and investigate the Fibonacci statistically convergent sequences and Fibonacci statistically Cauchy sequences. Using the definition of statistical boundedness, it will be proved that the set of Fibonacci statistically convergent sequence spaces of real numbers is a closed linear space of a set of Fibonacci bounded sequences of real numbers and nowhere dense in Fibonacci bounded sequences of real numbers. After proving that the set of Fibonacci statistically convergent sequences is dense in Frechet metric space of all real sequences, the inclusion relations will be given.
For the rest of the paper, firstly an approximation theorem, which is an analogue of Korovkin theorem, is given and an example is solved. Second, the rate of Fibonacci statistical convergence of a sequence of positive linear operators defined \(C_{2\pi}(\mathbb{R})\) into \(C_{2\pi}(\mathbb{R})\) is computed.
3 Main results
3.1 Fibonacci type statistical convergence
Lemma 3.1
If \(X\subset Y\), then \(X(\widehat{F}) \subset Y(\widehat{F})\).
Theorem 3.2
Consider that X is a Banach space and A is a closed subset of X. Then \(A(\widehat{F})\) is also closed in \(X(\widehat{F})\).
Proof
Since A is a closed subset of X, from Lemma 3.1, then we can write \(A(\widehat{F}) \subset X(\widehat{F})\). \(\overline{A( \widehat{F})}\), A̅ denote the closure of \(A(\widehat{F})\) and A, respectively. To prove the theorem, we must show that \(\overline{A(\widehat{F})}=\overline{A}(\widehat{F})\).
Conversely, if we take \(x\in\overline{A(\widehat{F})}\), then \(x\in A(\widehat{F})\). We know that A is closed. Then \(\overline{A( \widehat{F})}=\overline{A}(\widehat{F})\). Hence, \(A(\widehat{F})\) is a closed subset of \(X(\widehat{F})\). □
From this theorem, we can give the following corollary.
Corollary 3.3
If X is a separable space, then \(X(\widehat{F})\) is also a separable space.
Definition 3.4
In this case we write \(d(\widehat{F})-\lim x_{k}=L\) or \(x_{k}\rightarrow L(S(\widehat{F}))\). The set of F̂-statistically convergent sequences will be denoted by \(S(\widehat{F})\). In the case \(L=0\), we will write \(S_{0}(\widehat{F})\).
Definition 3.5
Theorem 3.6
If x is an F̂-statistically convergent sequence, then x is an F̂-statistically Cauchy sequence.
Proof
Let \(\varepsilon>0\). Assume that \(x_{k}\rightarrow L(S(\widehat{F}))\). Then \(\vert \widehat{F}x_{k}-L\vert <\varepsilon/ 2\) for almost all k. If N is chosen so that \(\vert \widehat{F}x_{N}-L\vert <\varepsilon/ 2\), then we have \(\vert \widehat{F}x_{k}-\widehat{F}x_{N}\vert < \vert \widehat{F}x_{k}-L\vert +\vert \widehat{F}x_{N}-L\vert <\varepsilon/ 2 + \varepsilon/ 2 =\varepsilon\) for almost all k. It means that x is an F̂-statistically Cauchy sequence. □
Theorem 3.7
If x is a sequence for which there is an F̂-statistically convergent sequence y such that \(\widehat{F}x_{k}=\widehat{F}y_{k}\) for almost all k, then x is an F̂- statistically convergent sequence.
Proof
Definition 3.8
[38]
Although a statistically convergent sequence does not need to be bounded (cf. [39, 40]), the following proposition shows that every statistically convergent sequence is statistically bounded.
Now, using Propositions 3.9 and 3.10, we can give the following corollary.
Corollary 3.11
Every F̂-statistically convergent sequence is F̂-statistically bounded.
Denote the set of all F̂-bounded sequences of real numbers by \(m(\widehat{F})\) [15]. Based on Definition 3.8 and the descriptions of \(m_{0}\) and \(m(\widehat{F})\), we can denote the set of all F̂-bounded statistically convergent sequences of real numbers by \(m_{0}(\widehat{F})\).
The following theorem can be proved by Theorem 2.1 of [41] and Theorem 3.2.
Theorem 3.12
The set of \(m_{0}(\widehat{F})\) is a closed linear space of the linear normed space \(m(\widehat{F})\).
Theorem 3.13
The set of \(m_{0}(\widehat{F})\) is a nowhere dense set in \(m( \widehat{F})\).
Proof
Theorem 3.14
The set of F̂-statistically convergent sequences is dense in the space ω.
Proof
If \(x=(x_{k})\in S(\widehat{F})\) (for all k) and the sequence \(y=(y_{k})\) (for all k) of real numbers differs from x only in a finite number of terms, then evidently \(y\in S(\widehat{F})\), too. From this statement the proof follows at once on the basis of the definition of the metric in ω. □
Theorem 3.15
- i.
The inclusion \(c(\widehat{F})\subset S(\widehat{F})\) is strict.
- ii.
\(S(\widehat{F})\) and \(\ell_{\infty}(\widehat{F})\) overlap but neither one contains the other.
- iii.
\(S(\widehat{F})\) and \(\ell_{\infty}\) overlap but neither one contains the other.
- iv.
S and \(S(\widehat{F})\) overlap but neither one contains the other.
- v.
S and \(c(\widehat{F})\) overlap but neither one contains the other.
- vi.
S and \(c_{0}(\widehat{F})\) overlap but neither one contains the other.
- vii.
S and \(\ell_{\infty}(\widehat{F})\) overlap but neither one contains the other.
Proof
For the other items, firstly, use the inclusion relations in [15]. It is obtained that the inclusions \(c\subset S( \widehat{F})\), \(c\subset c(\widehat{F})\), \(c\subset m(\widehat{F})\), \(c\subset S\), \(c\subset\ell_{\infty}\) and \(c\cap c_{0}(\widehat{F}) \neq\phi\) hold. Then we see that \(S(\widehat{F})\) and \(\ell_{\infty }(\widehat{F})\), \(S(\widehat{F})\) and \(\ell_{\infty}\), S and \(S(\widehat{F})\), S and \(c(\widehat{F})\), S and \(c_{0}( \widehat{F})\), S and \(\ell_{\infty}(\widehat{F})\) overlap.
ii) We define \(\widehat{F}x=\widehat{F}x_{n}\) by (3.1). Then \(\widehat{F}x\in S\), but F̂x is not in \(\ell_{\infty}\). Now we choose \(u=(1,0,1,0,\ldots)\). Then \(u\in\ell_{\infty}( \widehat{F})\) but \(u\notin S(\widehat{F})\).
iii) The proof is the same as (ii).
(v), (vi) and (vii) are proven similar to (iv). □
3.2 Approximation theorems
3.2.1 Approximation by F̂-statistical convergence
In this section, we get an analogue of classical Korovkin theorem by using the concept of F̂-statistical convergence.
Theorem 3.16
Theorem 3.17
Our main Korovkin type theorem is given as follows.
Theorem 3.18
Proof
We remark that our Theorem 3.18 is stronger than Theorem 3.17 as well as Theorem of Gadjiev and Orhan [17]. For this purpose, we get the following example.
Example 3.19
3.2.2 Rate of F̂-statistical convergence
In this section, we estimate the rate of F̂-statistical convergence of a sequence of positive linear operators defined by \(C_{2\pi}(\mathbb{R})\) into \(C_{2\pi}(\mathbb{R})\). Now, we give the following definition.
Definition 3.20
As usual we have the following auxiliary result.
Lemma 3.21
- (i)
\(\alpha(x_{k}-L_{1})=d(\widehat{F})-o(a_{n})\) for any scalar α,
- (ii)
\((x_{k}-L_{1})\pm(y_{k}-L_{2})=d(\widehat{F})-o(c_{n})\),
- (iii)
\((x_{k}-L_{1})(y_{k}-L_{2})=d(\widehat{F})-o(a_{n}b_{n})\),
Theorem 3.22
Proof
4 Conclusion
One of the most known and interesting number sequences is the Fibonacci sequence, and it still continues to be of interest to mathematicians because this sequence is an important and useful tool to expand the mathematical horizon for many mathematicians.
The concept of statistical convergence for a sequence of real numbers was defined by Fast [2] and Steinhaus [43] independently in 1951. Statistical convergence has recently become an area of active research. Currently, researchers in statistical convergence have devoted their effort to statistical approximation.
Approximation theory has important applications in the theory of polynomial approximation in various areas of functional analysis. The study of the Korovkin type approximation theory is a well-established area of research, which is concerned with the problem of approximating a function f by means of a sequence \(A_{n}\) of positive linear operators. Statistical convergence is quite effective in the approximation theory. In recent times, very high quality publications have been made using approximation theory and statistical convergence [9, 10, 18–24].
In this study, we have studied the concept of statistical convergence which has an important place in the literature using Fibonacci sequences. The statistical convergence is a generalization of the usual notion of convergence. We have defined the Fibonacci type statistical convergence and investigated basic properties. A new version of Korovkin type approximation theory was introduced using the new concept of statistical convergence.
Declarations
Acknowledgements
The first author was supported by Scientific Research Projects Coordination Unit of Istanbul University. Project number: 26287.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Freedman, AR, Sember, JJ: Densities and summability. Pac. J. Math. 95(2), 293-305 (1981) MathSciNetView ArticleMATHGoogle Scholar
- Fast, H: Sur la convergence statistique. Colloq. Math. 2, 241-244 (1951) MathSciNetView ArticleMATHGoogle Scholar
- Schoenberg, IJ: The integrability of certain functions and related summability methods. Am. Math. Mon. 66, 361-375 (1959) MathSciNetView ArticleMATHGoogle Scholar
- Buck, RC: Generalized asymptotic density. Am. J. Math. 75, 335-346 (1953) MathSciNetView ArticleMATHGoogle Scholar
- Zygmund, A: Trigonometric Series. Cambridge University Press, Cambridge (1979) MATHGoogle Scholar
- Erdos, R, Tenenbaum, G: Sur les densities de certaines suites d’entiers. Proc. Lond. Math. Soc. 3(59), 417-438 (1989) MathSciNetView ArticleMATHGoogle Scholar
- Miller, HI: A measure theoretical subsequence characterization of statistical convergence. Trans. Am. Math. Soc. 347, 1811-1819 (1995) MathSciNetView ArticleMATHGoogle Scholar
- Maddox, IJ: Statistical convergence in a locally convex sequence space. Math. Proc. Camb. Philos. Soc. 104, 141-145 (1988) View ArticleMATHGoogle Scholar
- Aktuğlu, H: Korovkin type approximation theorems proved via αβ-statistical convergence. J. Comput. Appl. Math. 259, 174-181 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Karakaya, V, Karaisa, A: Korovkin type approximation theorems for weighted \(\alpha \beta-\)statistical convergence. Bull. Math. Sci. 5, 159-169 (2015). doi:10.1007/s13373-015-0065-y MathSciNetView ArticleMATHGoogle Scholar
- Braha, NL: Some weighted equi-statistical convergence and Korovkin type-theorem. Results Math. 70(3), 433-446 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Fridy, JA: On statistical convergence. Analysis 5, 301-313 (1985) MathSciNetView ArticleMATHGoogle Scholar
- Koshy, T: Fibonacci and Lucas Numbers with Applications. Wiley, New York (2001) View ArticleMATHGoogle Scholar
- Goonatilake, S: Toward a Global Science p. 126. Indiana University Press, (1998) Google Scholar
- Kara, EE: Some topological and geometrical properties of new Banach sequence spaces. J. Inequal. Appl. 2013, 38 (2013). doi:10.1186/1029-242X-2013-38 MathSciNetView ArticleMATHGoogle Scholar
- Korovkin, PP: Linear Operators and Approximation Theory. Hindustan Publishing, New Delhi (1960) Google Scholar
- Gadjiev, AD, Orhan, C: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 32, 129-138 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Belen, C, Mohiuddine, SA: Generalized weighted statistical convergence and application. Appl. Math. Comput. 219(18), 9821-9826 (2013) MathSciNetMATHGoogle Scholar
- Edely, OHH, Mohiuddine, SA, Noman, AK: Korovkin type approximation theorems obtained through generalized statistical convergence. Appl. Math. Lett. 23(11), 1382-1387 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Edely, OHH, Mursaleen, M, Khan, A: Approximation for periodic functions via weighted statistical convergence. Appl. Math. Comput. 219(15), 8231-8236 (2013) MathSciNetMATHGoogle Scholar
- Mohiuddine, SA: An application of almost convergence in approximation theorems. Appl. Math. Lett. 24(11), 1856-1860 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen, M, Alotaibi, A: Statistical summability and approximation by de la Valle-Poussin mean. Appl. Math. Lett. 24(3), 320-324 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen, M, Alotaibi, A: Statistical lacunary summability and a Korovkin type approximation theorem. Ann. Univ. Ferrara 57, 373-381 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Mursaleen, M, Karakaya, V, Ertürk, M, Gürsoy, F: Weighted statistical convergence and its application to Korovkin type approximation theorem. Appl. Math. Comput. 218(18), 9132-9137 (2012) MathSciNetMATHGoogle Scholar
- Kara, EE, Başarır, M: An application of Fibonacci numbers into infinite Toeplitz matrices. Caspian J. Math. Sci. 1(1), 43-47 (2012) Google Scholar
- Alotaibi, A, Mursaleen, M, Alamri, BAS, Mohiuddine, SA: Compact operators on some Fibonacci difference sequence spaces. J. Inequal. Appl. 2015, 203 (2015). doi:10.1186/s13660-015-0713-5 MathSciNetView ArticleGoogle Scholar
- Başarır, M, Başar, F, Kara, EE: On the spaces of Fibonacci difference null and convergent sequences (2013). arXiv:1309.0150
- Candan, M: A new approach on the spaces of generalized Fibonacci difference null and convergent sequences. Math. Æterna 5(1), 191-210 (2015) Google Scholar
- Candan, M, Kılı nç, G: A different look for paranormed Riesz sequence space of derived by Fibonacci matrix. Konuralp J. Math. 3(2), 62-76 (2015) MathSciNetMATHGoogle Scholar
- Candan, M, Kara, EE: A study on topological and geometrical characteristics of new Banach sequence spaces. Gulf J. Math. 3(4), 67-84 (2015) MathSciNetMATHGoogle Scholar
- Candan, M, Kayaduman, K: Almost convergent sequence space derived by generalized Fibonacci matrix and Fibonacci core. Br. J. Math. Comput. Sci. 7(2), 150-167 (2015) View ArticleGoogle Scholar
- Demiriz, S, Kara, EE, Başarır, M: On the Fibonacci almost convergent sequence space and Fibonacci core. Kyungpook Math. J. 55, 355-372 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Kara, EE, Başarır, M, Mursaleen, M: Compactness of matrix operators on some sequence spaces derived by Fibonacci numbers. Kragujev. J. Math. 39(2), 217-230 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Kara, EE, Demiriz, S: Some new paranormed difference sequence spaces derived by Fibonacci numbers. Miskolc Math. Notes 16(2), 907-923 (2015). doi:10.18514/MMN.2015.1227 MathSciNetView ArticleMATHGoogle Scholar
- Kara, EE, Ilkhan, M: Some properties of generalized Fibonacci sequence spaces. Linear Multilinear Algebra 64(11), 2208-2223 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Uçar, E, Basar, F: Some geometric properties of the domain of the double band matrix defined by Fibonacci numbers in the sequence space \(\ell_{\infty}\). AIP Conf. Proc. 1611, 316-324 (2014). doi:10.1063/1.4893854 View ArticleGoogle Scholar
- Kreyszig, E: Introductory Functional Analysis with Applications. Wiley, New York (1978) MATHGoogle Scholar
- Fridy, JA, Orhan, C: Statistical limit superior and limit inferior. Proc. Am. Math. Soc. 125(12), 3625-3631 (1997) MathSciNetView ArticleMATHGoogle Scholar
- Bhardwaj, VK, Gupta, S: On some generalizations of statistical boundedness. J. Inequal. Appl. 2014, 12 (2014). doi:10.1186/1029-242X-2014-12 MathSciNetView ArticleMATHGoogle Scholar
- Fridy, JA, Demirci, K, Orhan, C: Multipliers and factorizations for bounded statistically convergent sequences. Analysis 22, 321-333 (2002) MathSciNetMATHGoogle Scholar
- Salat, T: On statistically convergent sequences of real numbers. Math. Slovaca 30, 139-150 (1980) MathSciNetMATHGoogle Scholar
- Gadziev, AD: The convergence problems for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P. Korovkin. Sov. Math. Dokl. 15, 1433-1436 (1974) MATHGoogle Scholar
- Steinhaus, H: Sur la convergence ordinaire et la convergence asymptotique. Colloq. Math. 2, 73-74 (1951) View ArticleGoogle Scholar