Conditions i and ii as above are the regularity conditions of the proposed deferred weighted mean [ 29 ]. Let p n be the sequence of non-negative real numbers such that. Definition 1 generalizes various known definitions as analyzed in Remark 1. Lastly, if. As a characterization of the deferred weighted regular methods, we present the following theorem.
Assume that 2. Thus, we have. Therefore, 2. Next, for statistical version, we present below the following definitions.
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In order to prove that the converse is not true, we present Example 2 below. We also consider the sequence x n by. In the last few decades, many researchers emphasized expanding or generalizing the Korovkin-type hypotheses from numerous points of view in light of a few distinct angles, containing for instance space of functions, Banach spaces summability theory, etc. Certainly, the change of Korovkin-type hypothesis is far from being finished till today. For additional points of interest and outcomes associated with the Korovkin-type hypothesis and other related advancements, we allude the reader to the current works [ 7 — 10 , 22 ], and [ 17 ].
Based upon the proposed methodology and techniques, we intend to estimate the rate of convergence and investigate the Korovkin-type approximation results. In fact, we extend here the result of Kadak et al.
Then we say that T is a positive linear operator provided. From equations 3. Clearly, from the above supposition for the implication in 3. Subsequently, we obtain. Hence, implication 3. Furthermore, if we substitute. We also observe that. It is clear that the sequence T n satisfies the conditions 3. We present the following definition. Also let u n be a positive non-decreasing sequence. Let u n and v n be two positive non - decreasing sequences. As assertions ii to iv of Lemma 1 are quite similar to i , so it can be proved along similar lines. Hence, the proof of Lemma 1 is completed.
Also let u n and v n be positive non - decreasing sequences. We assume that the following conditions i and ii are satisfied :.
Using 4. Lastly, for the sake of conditions i and ii of Theorem 3 in conjunction with Lemma 1 , inequalities 4. This completes the proof of Theorem 4. In this concluding section of our investigation, we present several further remarks and observations concerning various results which we have proved here. Then, since. However, since x n is not ordinarily convergent, it does not converge uniformly in the ordinary sense. Thus, for the operators defined in 3.
Hence, this application clearly indicates that our Theorem 3 non-trivially generalizes is stronger than the usual Korovkin-type theorem see [ 32 ]. Now, by applying 5. Thus, our Theorem 3 is also a non-trivial extension of Kadak et al. Based upon the above results, it is concluded here that our proposed method has successfully worked for the operators defined in 3.
We replace conditions i and ii in our Theorem 4 by the condition. It now follows from 5. Swardson, Strong integral summability and the Stone-Cech compactification of the half-line, Pacific J. Makarov, M. Levin, and A. Pehlivan and M. Maio, and L. Kocinac, Statistical convergence in topology, Topology Appl.
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Notifications View Subscribe. Abstract An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. Let P denote the space whose elements are finite sets of distinct positive integers. Keywords Ideal convergence, ideal continuity, -sequence, quasi-Cauchy sequence. Full Text: PDF.
References D. Monthly, , 4, , CakalliOnGcontinuity : H. Hazarika, On ideal convergence in topological groups, Scientia Magna,7 4 , G"ahler, 2-metrische R"aume and ihre topologische Struktur, Math. Keane : M. Keane, Understanding Ergodicity, Integers 11B Zygmund, Trigonometric Series, Cambridge Univ. Free Preview. Discusses the theory of classical and modern methods in summability Includes a technique for studying the existence of solutions of infinite systems of differential equations in Banach sequence spaces Introduces the approximation of functions by linear positive operators Highlights interesting connections between convergence methods and approximation results Presents original papers from active researchers around the globe see more benefits.
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FAQ Policy. One of the most active areas of research in summability theory is the concept of statistical convergence, which is a generalization of the familiar and widely investigated concept of convergence of real and complex sequences, and it has been used in Fourier analysis, probability theory, approximation theory and in other branches of mathematics.