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THE \lambda^{+r}(^\mue)-STATISTICAL CONVERGENCE
Abstract
Let λ = (λn)n≥1be a nondecreasing sequence of positive numbers tending to infinity such that λ1 = 1 and λn+1 ≤ λn + 1 for all n, and let In = [n - λn + 1; n] for n = 1; 2; . . . . Then for any given nonzero sequence μ, we define by Δ⁺(μ) the operator that generalizes the operator of the first difference and is defined by Δ+(μ)xk = μk(xk - xk+1). In this article, for any given integer r ≥ 1, we deal with the λ+r(μ)-statistical convergence that generalizes in a certain sense the well-known λrE-statistical convergence. The main results consist in determining sets of sequences χ and χ of the form s⁰ξ satisfying χ ⊂ [V; λ]0(Δ+r(μ)) ⊂ χ' and sets κ and κ' of the form sε satisfying κ ≤ [V; λ]∞(λ+r(μ)) ⊂ χ'. This study is justified since the infinite matrix associated with the operator Δ+r(μ) cannot be explicitly calculated for all r.
... In this study, the results from [3] were extended and some new results were obtained using Deferred Cesaro mean defined by [1] in as follows: ...
... (iv) q(n) = n and p(n) = n − λ n where (λ n ) is a nondecrasing sequence of natural numbers such that λ 1 = 1 and λ n+1 ≤ λ n + 1 holds then (1.3) coincides with the λ +r (µ)-statistical convergence defined by [3] and with the definition of λ m -statistcal convergence defined by [5]. ...
... for u ∈ C (see in [3] Lemma 2.2). Then, b = (b n ) ∈ [D q p ] α . ...
Let p(n) and q(n) be nondecreasing sequence of positive integers such that p(n) < q(n) and limn→∞ q(n) = ∞ holds. For any r ∈ Z^+, we define D_p,q^+r- statistical convergence of ∆^+r x where ∆^+r is r- th difference of the sequence (x_n). The main results in this paper consist in determining sets of sequences χ and χ' of the form [D_ p^q]_0 α satisfying χ ⊂ [D_p^q]_0(∆^+r ) ⊂ χ ' and sets φ and φ' of the form [D_p^q]_α satisfying φ ≤ [D_p^q]_∞(∆^+r ) ≤ φ' .
... Subsequently, many important modifications of this concept were studied. For example,statistical convergence was introduced in (Mursaleen, 2000) and ideal convergence was given in Kostyrko et al., 2000. Later on, Savas and Das (2011) defined --statistical convergence. ...
... Proof It can be easily proved by assuming that there are two and making a contradiction. Now, let's recall the definition of the filter (Mursaleen, 2000). Let ≠ ∅ and a family of ℱ ⊂ ( ) is a filter where, ∅ ∉ ℱ; for all 1 , 2 ∈ ℱ then 1 ∩ 2 ∈ ℱ; for all 1 ∈ ℱ and 1 ⊂ 2 then 2 ∈ ℱ. ...
In this paper, λI-statistical convergence is defined to generalize statistical convergence on Neutrosophic normed spaces. As it is known, Neutrosophic theory, which brings a new breath to daily life and complex scientific studies which we encounter with many uncertainties, is a rapidly developing field with many new study subjects.
Thus, researchers show great interest in this philosophical approach and try to transfer related topics to this field quickly. For this purpose, in this study, besides the definition of λI-statistical convergence, the important features of Hilbert sequence space and λI-statistical convergence in Neutrosophic spaces are examined with the help of these defined sequences. By giving the relationship between Hilbert λI-statistical convergence and Hilbert I-statistical convergence, it has been evaluated whether the definitions contain a coverage relationship as in fuzzy and intuitionistic fuzzy.
As a result, it is thought that the selected convergence type is suitable for the Neutrosophic normed space structure and is a guide for new convergence types.
... is turned to Lacunary Statistical convergence[9], (iii) If q(n) = λ n and p(n) = 0 (where λ n is a strictly increasing sequence of natural numbers such that lim n λ n = ∞), then Definition 1.1. is coincide λ−statistical convergence of sequences which is given by Osikievich[18]and Mursaleen[13]. ...
In this paper, deferred statistical convergence is defined by using deferred Cesàro mean instead of Cesàro mean in the definition of statistical convergence. The obtained method is compared with strong deferred Cesàro mean and statistical convergence under some certain assumptions. Also, some inclusion theorems and examples are given.
The concepts of —summability, —strong harmonic summability (denoted by ])—statistical logarithmic convergence (denoted by ]) and —statistically logarithmic boundedness of sequences of real numbers are introduced and tried to investigate some relations between the —strongly harmonically summability and —statistical logarithmic convergence in this work. We also establish some connections between and . It is shown that if a sequence is bounded then it is —statistically logarithmic boundedness. In addition, we define —harmonic summability and—statistical logarithmic convergence of real number sequences and provide some inclusion connections between these spaces.
In this research paper, our aim is to introduce the concepts of -statistical convergence of order and -statistical boundedness of order by using unbounded modulus functions in metric spaces. Furthermore, we obtain some relations between these concepts, and we provide some counterexamples to support our results.
Recently, Devaiya and Srivastava [3] studied the -strong convergence of numerical sequences and Fourier series using a lower triangular matrix , and generalized the results of Kórus [8]. The main objective of this paper is to introduce -strongly convergent sequence spaces for , and defined by a sequence of modulus functions. We also provide a relationship between [T,G,u,q] and -strongly convergent sequence spaces. Further, we investigate some geometrical and topological characteristics and establish some inclusion relationships between these sequence spaces. In the last, we derive some results on characterizations for -strong convergent sequences, statistical convergence and Fourier series using the idea of -strongly convergent sequence spaces.
We investigate the concept of -strongly summable and -statistical convergence for double sequences of order in topological groups by using definition of -statistical convergence which was defined by Cakalli in Bulletin of the Malaysian Mathematical Sciences Society 40:1701–1710 . We give some inclusion relations between these concepts.
In this study, we introduce and examine statistically summability, weighted statistical convergence and -summability and some inclusion relations between them. In addition, we define -summability with respect to an Orlicz function and examine some prpoperties of this new concept.
In this paper, we introduced the space of all deferred -statistically convergent sequences of order in the normed space (X, q) with the help of generalized difference operator , unbounded modulus function f, and deferred sequences of non-negative integers with for any , . We also introduced the space of all strongly deferred -summable sequences of order in the normed space (X, q). Inclusion relations between spaces and are established under certain conditions.KeywordsStatistical convergenceModulus functionDifference sequence spaceDeferred sequences
Our aim is to give concepts of density, statistical convergence, statistically Cauchy sequence, Cesàro summability and statistical convergence on non-Newtonian calculus. Especially, if a generator on this calculus is determined as identity function, all results in the classical case are reached.
In this paper, we introduce the opinion of -almost statistical convergence of weight where for any sequence in with We also examine some relations.KeywordsWeight function rStatistical convergenceAlmost convergence
In this article, the we establish the generalized Ulam-Hyers stability of a generalized mixed Order Functional Equation in various Banach Spaces.
The idea of statistical convergence was given by Zygmund [45] in the first edition of his monograph published in Warsaw in 1935 and then was considered as a summability method by Schoenberg [38] in 1959 despite the fact that it made its initial appearance in a short note by Fast [16] and Steinhaus [42] in 1951. Along with the theory of summability it has played an important role in Fourier analysis, Ergodic theory, Number theory, Measure theory, Trigonometric series, Turnpike theory and Banach spaces. Salat [35] also showed that the set of bounded statistical convergent real valued sequences is a closed subspace of bounded sequences in 1980. After that Fridy [17] introduced the concept of statistically Cauchiness of sequences and proved that it is equivalent to statistical convergence. More importantly he proved that there is no any matrix summability method which involves statistical convergence in the same paper in 1985. Another magnificent development was presented to the literature by Connor [8] in 1988. For the first time in the literature his work confirmed the direct link between statistical convergence and strong s-Cesàro summability by unveiling that the notions are equivalent for bounded sequences, where 0
In this paper, we examine and study statistical convergence of order of sequences of fuzzy numbers such that strong summability of order of sequences of fuzzy numbers such that and construct some interesting examples. We also give relations between these concepts. Furthermore, some relations between the space and are examined. The modulus function presents the containment connection.
In this study, by using definition of rho-statistical convergence which was defined by Cakalli [5], we give some inclusion relations between the sets of rho-statistical convergence and statistical convergence in topological groups.
In this paper we introduce - statistically convergent of order . In addition, we establish some connections between - statistically convergent of order and the – strong convergence of order .
In the present paper, we introduce the notion of λm-convergence and λm -boundedness. We examine the relationship of these notions between of the ordinary convergence and ordinary boundedness, respectively. We also introduce new sequence spaces and some topological properties of these spaces. Furthermore, we investigate some inclusion relations between these sequence spaces and other well known sequence spaces.
In this study, we examine the notion of Δλ-statistical boundedness on an arbitrary time scale. Moreover, some inclusion relations are obtained.
We introduce the notions of weighted lacunary statistical pointwise and uniform convergence and a kind of convergence which is lying between aforementioned convergence methods, namely, weighted lacunary equi-statistical convergence and obtain various implication results with supporting examples. We then apply our new concept of weighted lacunary equi-statistical convergence with a view to proving Korovkin and Voronovskaya type approximation theorems. We also construct an example with the help of generating functions type Meyer-König and Zeller which shows that our Korovkin-type theorem is stronger than its classical version. Moreover, we compute the rate of weighted lacunary equi-statistical convergence for operators in terms of modulus of continuity.
In this paper, we introduce new notion, namely, double statistical convergence in topological groups. We mainly investigate some inclusion relations between double statistical and double statistical convergence.
In this paper, we introduce and study statistical convergence, statistical boundedness and strong summability of sequences in metric spaces. Furthermore we establish some relations between the sets of -statistically convergent sequences, between the sets of -statistically bounded sequences, between the sets of statistical convergent sequences and the sets of strongly summable sequences for various sequences in Furthermore we establish some inclusion relations between the sets of strongly summable sequences for various sequences in set .
Recently, Kórus [7] studied the -strong convergence of numerical sequences. By using the idea of this paper, we introduce -strongly convergent sequence spaces defined by a sequence of modulus functions. We also make an effort to study some inclusion relations, topological and geometric properties of these spaces. Some characterizations for strong convergent sequences are given. Finally, we study statistical convergence over these spaces and problems related to Fourier series.
A sequence of points in , the set of real numbers, is called
-statistically convergent to an element of if for each , where is a non-decreasing sequence of positive real numbers tending to such that , , and for each positive integer n. A real-valued function defined on a subset of is called -statistically ward continuous if it preserves -statistical quasi-Cauchy sequences where a sequence is defined to be -statistically quasi-Cauchy if the sequence is -statistically convergent to 0. We obtain results related to -statistical ward continuity, -statistical ward compactness, ward continuity, continuity, and uniform continuity. It turns out that the set of uniformly continuous functions coincides with the set of -statistically ward continuous functions not only on a bounded subset of , but also on an interval.
In this paper, we introduce the concept of λ-statistical convergence of order θ and strong λ-summability of order θ for the sequence of fuzzy numbers. Further the same concept is extended to the sequence of fuzzy functions and introduce the spaces like (Formula presented) and (Formula presented). Some inclusion relations in those spaces and also the underlying relation between these two spaces are also obtained.
In this paper, following the works of Phu (Numer. Funct. Anal. Optimiz., 2001), Aytar (Numer. Funct. Anal. Optimiz., 2008) and Ghosal (Application of Math., 2013) we make a new approach to extend the application area of rough statistical convergence usually used in sequence of real numbers to the theory of probability distributions. The introduction of these concepts in probability of rough statistical convergence, rough strong Ces\grave{\mbox{a}}ro summable, rough lacunary statistical convergence, rough -convergence, rough -statistical convergence and rough strong -summable generalize the convergence analysis to accommodate any form of distribution of random variables. Among these six concepts in probability only three convergences are distinct- rough statistical convergence, rough lacunary statistical convergence and rough -statistical convergence where rough strong Ces\grave{\mbox{a}}ro summable is equivalent to rough statistical convergence, rough -convergence is equivalent to rough lacunary statistical convergence, rough strong -summable is equivalent to rough -statistical convergence. Basic properties and interrelations of above mentioned three distinct convergences are investigated and make some observations about these classes and in this way we show that rough statistical convergence in probability is the more generalized concept than usual rough statistical convergence.
In this paper we define the sequence space by using an Orlicz
function and we study various properties and obtain some inclusion relations
involving this space. We give some relations between lacunary
statistical convergence and strongly lacunary convergence.
In this paper, we prove the Hyers–Ulam stability of an additive–quadratic–cubic–quartic functional equation in matrix normed spaces.
We construct the linear positive operators generated by the q-Dunkl generalization of the exponential function. We have approximation properties of the operators via a universal Korovkin-type theorem and a weighted Korovkin-type theorem. The rate of convergence of the operators for functions belonging to the Lipschitz class is presented. We obtain the rate of convergence by means of the classical, second order, and weighted modulus of continuity, respectively, as well.
In this paper, we introduce the concept Ŝαλ – statistical convergence of order α. Also some relations between Ŝαλ – statistical convergence of order α and strong ŵαp(λ) – summability of order α are given. Furthermore some relations between the spaces ŵα(p)[λ, M] – and Ŝαλ – are examined. © 2014, Eduem - Editora da Universidade Estadual de Maringa. All rights reserved.
In this paper, we introduce two symmetric q-Dunkl-classical orthogonal polynomial sets generalizing the Hermite polynomials as range of discrete q-Hermite polynomials I and II by a suitable orthogonality preserving operator. Then, we state some of their properties: explicit expressions, recurrence relations, generating functions, inversion formulas, linear functionals and components. We conclude by locating these polynomials among orthogonal polynomials generalizing Hermite polynomials and classified according to lowering operators.
We study statistical approximation properties of
𝑞
-Bernstein-Shurer operators and establish some direct theorems. Furthermore, we compute error estimation and show graphically the convergence for a function
𝑓
by operators and give its algorithm.
We introduce the Szasz-Kantorovitch-Bezier operators Sn,a which is the modified form of Szasz-Kantorovitch operators and study the rate of convergence of bounded variation functions for these operators.
In the present paper we introduce and investigate weighted statistical approximation properties of a q-analogue of the Baskakov and Baskakov-Kantorovich operators. By using a weighted modulus of smoothness, we give some direct
estimations for error in the case 0 < q < 1.
Keywordsq-integers-q-Baskakov operators-q-Baskakov-Kantorovich operators-Weighted space-Weighted modulus of smoothness
MSC41A36-41A30-41A25
The purpose of the paper is to introduce Stancu-type linear positive operators generated by Dunkl generalization of exponential function. We present approximation properties with the help of well-known Korovkin-type theorem and weighted Korovkin-type theorem and also acquire the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K-functional, and second-order modulus of continuity by Dunkl analogue of Szász operators. Copyright © 2015 John Wiley & Sons, Ltd.
The approximation of functions by linear positive operators is an important research topic in general mathematics and it also provides powerful tools to application areas such as computer-aided geometric design, numerical analysis, and solutions of differential equations. q-Calculus is a generalization of many subjects, such as hypergeometric series, complex analysis, and particle physics. This monograph is an introduction to combining approximation theory and q-Calculus with applications, by using well- known operators. The presentation is systematic and the authors include a brief summary of the notations and basic definitions of q-calculus before delving into more advanced material. The many applications of q-calculus in the theory of approximation, especially on various operators, which includes convergence of operators to functions in real and complex domain? forms the gist of the book. This book is suitable for researchers and students in mathematics, physics and engineering, and for professionals who would enjoy exploring the host of mathematical techniques and ideas that are collected and discussed in the book. © Springer Science+Business Media New York 2013. All rights are reserved.
The main purpose of this paper is to construct linear positive operators generated by Dunkl generalization of exponential function. We present qualitative result with the help of universal Korovkin-type theorem and weighted Korovkin-type theorem and also get quantitative estimates in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre’s K-functional, second order modulus of continuity and weighted modulus of continuity by Dunkl analogue of Szasz operators.
The main object of this paper is to prove a Korovkin type theorem for the test functions 1, cosx,sinx in the space C2π(R)C2π(R) of all continuous 2π2π-periodic functions on the real line RR. Our analysis is based upon the statistical summability involving the idea of the generalized de la Vallée Poussin mean. We also investigate the rate of the de la Vallée Poussin statistical summability of positive linear operators in the space C2π(R)C2π(R). Finally, we provide an interesting illustrative example in support of our result.
The idea of dierence sequence spaces was intro- duced by Kzmaz (9) and generalized by Et and C olak (6). In this paper we introduce the sequence spaces (V; ; f; p)0 ( r ; E), (V; ; f; p)1 ( r ; E), (V; ; f; p)1 ( r ; E), S ( r; E) and S 0( r ; E), where E is any Banach space, examine them and give var- ious properties and inclusion relations on these spaces. We also show that the space S ( r; E) may be represented as a (V; ; f; p)1 ( r ; E) space.
The paper studies the convergence of P(u, x) to f(x) as u —> °o . The results obtained are generalized analogs, for the interval 0< x < °°, of known properties of S. Bernstein's approximation polynomials in a finite interval. 1. With a function j(t) in the closed interval (0,1), S. Bernstein in 1912 associated the poly- nomials
In this work, giving a modification of the well-known Szasz-Mirakjan operators, we prove that the error estimation of our operators is better than that of the classical Szasz-Mirakjan operators. Furthermore, we obtain a Voronovskaya type theorem for these modified operators. (c) 2007 Elsevier Ltd. All rights reserved.
In this paper, we present a modification of a sequence of mixed summation-integral type operators having Szasz and Beta basis functions in summation and integration, the so-called Szasz-Mirakjan-Beta operators. Then we show that our modified operators have a better error estimation on the interval [0, 2]. Furthermore, we give an r-th order generalization of the modified Szasz-Mirakjan-Beta operators and investigate their approximation properties.
In this paper, we introduce a modification of the Szasz-Mirakjan-Kantorovich operators, which preserve the linear functions. This type of operator modification enables better error estimation on the interval [1/2, +infinity) than the classical Szasz-Mirakjan-Kantorovich operators. We also obtain a Voronovskaya-type theorem for these operators.
A sequence of positive linear operators which approximate each continuous function on [0,1] while preserving the functione
2
(x) =x
2 is presented. Quantitative estimates are given and are compared with estimates of approximation by Bernstein polynomials.
Connections with summability are discussed.
In the present paper we propose q-analogue of the well known Szász-Kantorovich operators. We study local approximation as well as weighted approximation properties
of these new operators.
Keywords
q-Szász-Kantorovich operators–
q integers–
q Gamma function–Modulus of continuity–Approximation
The concept of λ-statistical convergence was introduced in [M. Mursaleen, λ-statistical convergence, Math Slovaca, 50 (2000) 111–115] by using the generalized de la Vallée Poussin mean. In this work we apply this method to prove some Korovkin type approximation theorems.
In the present paper, we introduce a Kantorovich type modification of q-Szász–Mirakjan operators and obtain weighted statistical approximation properties of these operators. Also for introduced operators, we give a Voronovskaja type theorem related to q-derivatives.
In this paper, we introduce a q-analogue of the Szász–Mirakyan–Kantorovich operators and we propose two different modifications of the q-Szász–Mirakyan–Kantorovich operators. These modifications preserve some test functions. We also examine the rate of convergence for the constructed operators by means of modulus of continuity.
The concept of almost convergence was introduced by Lorentz in 1948 [G.G. Lorentz, A contribution to theory of divergent sequences, Acta Math. 80 (1948) 167–190], and has various applications. In this work we apply this method to prove some Korovkin type approximation theorems.
On the q-definite integrals
- F H Jackson
Jackson, FH: On the q-definite integrals. Pure Appl. Math. Q. 41, 193-203 (1910)
A: A q-analogue of the Bernstein operator In: Seminar on Numerical and Statistical Calculus
- Lupas
Lupas, A: A q-analogue of the Bernstein operator. In: Seminar on Numerical and Statistical Calculus, University of
Cluj-Napoca, Cluj-Napoca, vol. 9, pp. 85-92 (1987)
- Sci
- Urss
Sci. URSS (NS) 31, 201-205 (1941)