In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p...In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings.展开更多
The sequences defined in Example 3 and Example 4 do not serve our purpose for any λ = （λn）. Because this sequences are just the sequences x = （xk） = （k） and x = （xk） = （1） respectively and any term of thes...The sequences defined in Example 3 and Example 4 do not serve our purpose for any λ = （λn）. Because this sequences are just the sequences x = （xk） = （k） and x = （xk） = （1） respectively and any term of these sequences can not be 0. In this short not we give Example 3＊ and Example 4＊ to show that the inclusions given in Theorem 2.4 and Theorem 2.9 are strict for some λ = （λn） , α and β such that 0 α β ≤ 1.展开更多
In this article we introduce the difference sequence spaces Wo [f, △m], W1 [f, △m],W∞[f ,△m] and S[f, △m], defined by a modulus function f. We obtain a relation between W1 【f, △m] ∩ l∞[f, △m] and S[f, △m]...In this article we introduce the difference sequence spaces Wo [f, △m], W1 [f, △m],W∞[f ,△m] and S[f, △m], defined by a modulus function f. We obtain a relation between W1 【f, △m] ∩ l∞[f, △m] and S[f, △m]∩ l∞[f, △m] and prove some inclusion results.展开更多
In this article, we introduce the concept of lacunary statistical convergence of order a of real number sequences and give some inclusion relations between the sets of lacu- nary statistical convergence of order α an...In this article, we introduce the concept of lacunary statistical convergence of order a of real number sequences and give some inclusion relations between the sets of lacu- nary statistical convergence of order α and strong Nα （p）-summability. Furthermore, some relations between the spaces Nθα （p） and Sθα are examined.展开更多
In this paper,we introduce the concept of λ-statistical convergence of order α.Also some relations between the λ-statistical convergence of order α and strong（V,λ）-summability of order α are given.
The paper aims to investigate different types of weighted ideal statistical convergence and strongly weighted ideal convergence of double sequences of fuzzy numbers. Relations connecting ideal statistical convergence ...The paper aims to investigate different types of weighted ideal statistical convergence and strongly weighted ideal convergence of double sequences of fuzzy numbers. Relations connecting ideal statistical convergence and strongly ideal convergence have been investigated in the environment of the newly defined classes of double sequences of fuzzy numbers. At the same time, we have examined relevant inclusion relations concerning weighted (<i>λ</i>, <i>μ</i>)-ideal statistical convergence and strongly weighted (<i>λ</i>, <i>μ</i>)-ideal convergence of double sequences of fuzzy numbers. Also, some properties of these new sequence spaces are investigated.展开更多
In this paper, we introduce and study the concept of lacunary invariant convergence for sequences of sets with respect to modulus functionfand give some inclusion relations.
The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical conve...The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra of all subsets of N, and proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set of all such measures endowed with the topology of point-wise convergence on forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N. Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures.展开更多
The purpose of this paper is to discuss those kinds of statistical convergence, in terms of F , or ideal Z-convergence, which are equivalent to measure convergence defined by a single statistical measure. We prove a n...The purpose of this paper is to discuss those kinds of statistical convergence, in terms of F , or ideal Z-convergence, which are equivalent to measure convergence defined by a single statistical measure. We prove a number of characterizations of a single statistical measure μ-convergence by using properties of its corresponding quotient Banach space l∞/l∞ （Iμ）. We also show that the usual sequential convergence is not equivalent to a single measure convergence.展开更多
In this paper we extend the notion of A-statistical convergence to the （λ,μ）statistical convergence for double sequences x =（xjk）. We also determine some matrix transformations and establish some core theorems r...In this paper we extend the notion of A-statistical convergence to the （λ,μ）statistical convergence for double sequences x =（xjk）. We also determine some matrix transformations and establish some core theorems related to our new space of double sequences Sλ,μ.展开更多
In this paper we obtain a new version of the Orlicz-Pettis theorem by using statistical convergence. To obtain this result we prove a theorem of uniform convergence on matrices related to the statistical convergence.
In article, I present a study on upper and lower statistical convergence, and upper and lower strong fractional weighted mean convergence by moduli for triple sequences. One of the generalizations of the discrete oper...In article, I present a study on upper and lower statistical convergence, and upper and lower strong fractional weighted mean convergence by moduli for triple sequences. One of the generalizations of the discrete operator Cesàro, was weighted mean operators, which are linear operators, too. Given a modulus function f, I established that a triple sequence that is f-upper or lower strong fractional weighted mean convergent, in some supplementary conditions, is also f-lower or upper statistically convergent. The results of this paper adapt the results obtained in [1] and [2] to upper and lower strong fractional weighted mean convergence and to triple sequence concept. Furthermore, new concepts can be applied to the approximation theory, topology, Fourier analysis, analysis interdisciplinary with applications electrical engineering, robotics and other domains.展开更多
The author shows a characterization of a （unbounded） weakly filter convergent sequence which is parallel to that every weakly null sequence （xn） in a Banach space admits a norm null sequence （yn） with yn ∈ co...The author shows a characterization of a （unbounded） weakly filter convergent sequence which is parallel to that every weakly null sequence （xn） in a Banach space admits a norm null sequence （yn） with yn ∈ co（xk）k≥n for all n ∈ N. A version of the Radon-Riesz type theorem is also proved within the frame of the filter convergence.展开更多
In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and...In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and give some correlation theorem. It is shown that <em>x</em> = {<em>x<sub>ijk</sub></em>} is ideal statistically pre-Cauchy if and only if <img src="Edit_0f9eaa1e-440b-4bac-8bae-c50bb5e5c244.bmp" alt="" /> <em>D </em>(<em>x<sub>ijk</sub></em>, <em>x<sub>pqr</sub></em>) ≥ <em>ε</em>, <em>i</em> ≤ <em>m</em>,<em> j </em>≤ <em>n</em>, <em>t </em>≤ <em>k</em>}| ≥ <em>δ</em>} ∈<em>I</em>. At the same time, we have proved <em>x</em> = {<em>x<sub>ijk</sub></em>} is ideal statistically convergent to <em>x</em><sub>0</sub> if and only if <img src="Edit_343f4dfc-82c3-4985-aebc-95c52795bb2f.bmp" alt="" />. Also, some properties of these new sequence spaces are investigated.展开更多
Let I 2N be an ideal and let XI = span{χI ： I ∈ I}, and let pI be the quotient norm of l∞/XI. In this paper, we show first that for each proper ideal I 2N, the ideal convergence deduced by I is equivalent to p...Let I 2N be an ideal and let XI = span{χI ： I ∈ I}, and let pI be the quotient norm of l∞/XI. In this paper, we show first that for each proper ideal I 2N, the ideal convergence deduced by I is equivalent to pI-kernel convergence. In addition, let K = {x＊oχ（·） ： x＊∈ p（e）}, where p（x） = lim supn→∞1/n（∑k=1n｜x（k）｜, and let Iμ = {A N ： μ（A） = 0} for all μ = x＊oχ（·） ∈ K. Then Iμ is a proper ideal. We also show that the ideal convergence deduced by the proper ideal Iμ, the p-kernel convergence and the statistical convergence are also equivalent.展开更多
In this article we define the notion of statistically convergent difference double sequence spaces. We examine the spaces 2l∞（△, q）, 2c（△, q）, 2cB（△, q）, 2cR（△, q）, 2cBR（△,q） etc. being symmetric, soli...In this article we define the notion of statistically convergent difference double sequence spaces. We examine the spaces 2l∞（△, q）, 2c（△, q）, 2cB（△, q）, 2cR（△, q）, 2cBR（△,q） etc. being symmetric, solid, monotone, etc. We prove some inclusion results too.展开更多
The notion of ideal convergence is a generalization of statistical convecgence which has been intensively investigated in last few years. For an admissible ideal ∮ C N × N, the aim of the present paper is to int...The notion of ideal convergence is a generalization of statistical convecgence which has been intensively investigated in last few years. For an admissible ideal ∮ C N × N, the aim of the present paper is to introduce the concepts of ∮-convergence and ：∮＊-convergence for double sequences on probabilistic normed spaces （PN spaces for short）. We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide. We also define ∮-Cauchy and ：∮＊- Cauchy double sequences on PN spaces and show that ∮-convergent double sequences are ∮-Cauchy on these spaces. We establish example which shows that our method of convergence for double sequences on PN spaces is more general.展开更多
文摘In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings.
文摘The sequences defined in Example 3 and Example 4 do not serve our purpose for any λ = （λn）. Because this sequences are just the sequences x = （xk） = （k） and x = （xk） = （1） respectively and any term of these sequences can not be 0. In this short not we give Example 3＊ and Example 4＊ to show that the inclusions given in Theorem 2.4 and Theorem 2.9 are strict for some λ = （λn） , α and β such that 0 α β ≤ 1.
文摘In this article we introduce the difference sequence spaces Wo [f, △m], W1 [f, △m],W∞[f ,△m] and S[f, △m], defined by a modulus function f. We obtain a relation between W1 【f, △m] ∩ l∞[f, △m] and S[f, △m]∩ l∞[f, △m] and prove some inclusion results.
文摘In this article, we introduce the concept of lacunary statistical convergence of order a of real number sequences and give some inclusion relations between the sets of lacu- nary statistical convergence of order α and strong Nα （p）-summability. Furthermore, some relations between the spaces Nθα （p） and Sθα are examined.
文摘In this paper,we introduce the concept of λ-statistical convergence of order α.Also some relations between the λ-statistical convergence of order α and strong（V,λ）-summability of order α are given.
文摘The paper aims to investigate different types of weighted ideal statistical convergence and strongly weighted ideal convergence of double sequences of fuzzy numbers. Relations connecting ideal statistical convergence and strongly ideal convergence have been investigated in the environment of the newly defined classes of double sequences of fuzzy numbers. At the same time, we have examined relevant inclusion relations concerning weighted (<i>λ</i>, <i>μ</i>)-ideal statistical convergence and strongly weighted (<i>λ</i>, <i>μ</i>)-ideal convergence of double sequences of fuzzy numbers. Also, some properties of these new sequence spaces are investigated.
文摘In this paper, we introduce and study the concept of lacunary invariant convergence for sequences of sets with respect to modulus functionfand give some inclusion relations.
基金supported by the National Natural Science Foundation of China (Grant Nos. 10771175, 10471114)
文摘The question of establishing measure theory for statistical convergence has been moving closer to center stage, since a kind of reasonable theory is not only fundamental for unifying various kinds of statistical convergence, but also a bridge linking the studies of statistical convergence across measure theory, integration theory, probability and statistics. For this reason, this paper, in terms of subdifferential, first shows a representation theorem for all finitely additive probability measures defined on the σ-algebra of all subsets of N, and proves that every such measure can be uniquely decomposed into a convex combination of a countably additive probability measure and a statistical measure (i.e. a finitely additive probability measure μ with μ(k) = 0 for all singletons {k}). This paper also shows that classical statistical measures have many nice properties, such as: The set of all such measures endowed with the topology of point-wise convergence on forms a compact convex Hausdorff space; every classical statistical measure is of continuity type (hence, atomless), and every specific class of statistical measures fits a complementation minimax rule for every subset in N. Finally, this paper shows that every kind of statistical convergence can be unified in convergence of statistical measures.
文摘The purpose of this paper is to discuss those kinds of statistical convergence, in terms of F , or ideal Z-convergence, which are equivalent to measure convergence defined by a single statistical measure. We prove a number of characterizations of a single statistical measure μ-convergence by using properties of its corresponding quotient Banach space l∞/l∞ （Iμ）. We also show that the usual sequential convergence is not equivalent to a single measure convergence.
基金supported by the Department of Science and Technology,New Delhi (Grant No.SR/S4/MS:505/07)
文摘In this paper we extend the notion of A-statistical convergence to the （λ,μ）statistical convergence for double sequences x =（xjk）. We also determine some matrix transformations and establish some core theorems related to our new space of double sequences Sλ,μ.
基金Supported by Junta de Andalucia grant FQM 257supported by MEC Project MTM-2006-15546-C02-01
文摘In this paper we obtain a new version of the Orlicz-Pettis theorem by using statistical convergence. To obtain this result we prove a theorem of uniform convergence on matrices related to the statistical convergence.
文摘In article, I present a study on upper and lower statistical convergence, and upper and lower strong fractional weighted mean convergence by moduli for triple sequences. One of the generalizations of the discrete operator Cesàro, was weighted mean operators, which are linear operators, too. Given a modulus function f, I established that a triple sequence that is f-upper or lower strong fractional weighted mean convergent, in some supplementary conditions, is also f-lower or upper statistically convergent. The results of this paper adapt the results obtained in [1] and [2] to upper and lower strong fractional weighted mean convergence and to triple sequence concept. Furthermore, new concepts can be applied to the approximation theory, topology, Fourier analysis, analysis interdisciplinary with applications electrical engineering, robotics and other domains.
基金partially supported by the Natural Science Foundation of China(11426061,11501108)the Natural Science Foundation of Fujian province(2015J01579)
文摘The author shows a characterization of a （unbounded） weakly filter convergent sequence which is parallel to that every weakly null sequence （xn） in a Banach space admits a norm null sequence （yn） with yn ∈ co（xk）k≥n for all n ∈ N. A version of the Radon-Riesz type theorem is also proved within the frame of the filter convergence.
文摘In this paper, we extend the notions of ideal statistically convergence for sequence of fuzzy number. We introduce the notions ideal statistically pre-Cauchy triple sequences of fuzzy number about Orlicz function, and give some correlation theorem. It is shown that <em>x</em> = {<em>x<sub>ijk</sub></em>} is ideal statistically pre-Cauchy if and only if <img src="Edit_0f9eaa1e-440b-4bac-8bae-c50bb5e5c244.bmp" alt="" /> <em>D </em>(<em>x<sub>ijk</sub></em>, <em>x<sub>pqr</sub></em>) ≥ <em>ε</em>, <em>i</em> ≤ <em>m</em>,<em> j </em>≤ <em>n</em>, <em>t </em>≤ <em>k</em>}| ≥ <em>δ</em>} ∈<em>I</em>. At the same time, we have proved <em>x</em> = {<em>x<sub>ijk</sub></em>} is ideal statistically convergent to <em>x</em><sub>0</sub> if and only if <img src="Edit_343f4dfc-82c3-4985-aebc-95c52795bb2f.bmp" alt="" />. Also, some properties of these new sequence spaces are investigated.
基金supported by Plan Project of Education Department of Fujian Province(Grant No.JA11275)
文摘Let I 2N be an ideal and let XI = span{χI ： I ∈ I}, and let pI be the quotient norm of l∞/XI. In this paper, we show first that for each proper ideal I 2N, the ideal convergence deduced by I is equivalent to pI-kernel convergence. In addition, let K = {x＊oχ（·） ： x＊∈ p（e）}, where p（x） = lim supn→∞1/n（∑k=1n｜x（k）｜, and let Iμ = {A N ： μ（A） = 0} for all μ = x＊oχ（·） ∈ K. Then Iμ is a proper ideal. We also show that the ideal convergence deduced by the proper ideal Iμ, the p-kernel convergence and the statistical convergence are also equivalent.
文摘In this article we define the notion of statistically convergent difference double sequence spaces. We examine the spaces 2l∞（△, q）, 2c（△, q）, 2cB（△, q）, 2cR（△, q）, 2cBR（△,q） etc. being symmetric, solid, monotone, etc. We prove some inclusion results too.
文摘The notion of ideal convergence is a generalization of statistical convecgence which has been intensively investigated in last few years. For an admissible ideal ∮ C N × N, the aim of the present paper is to introduce the concepts of ∮-convergence and ：∮＊-convergence for double sequences on probabilistic normed spaces （PN spaces for short）. We give some relations related to these notions and find condition on the ideal ∮ for which both the notions coincide. We also define ∮-Cauchy and ：∮＊- Cauchy double sequences on PN spaces and show that ∮-convergent double sequences are ∮-Cauchy on these spaces. We establish example which shows that our method of convergence for double sequences on PN spaces is more general.
