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Öğe f-Statistical convergence on topological modules(Amer Inst Mathematical Sciences-Aims, 2022) Javier Garcia-Pacheco, Francisco; Kama, RamazanThe classical notion of statistical convergence has recently been transported to the scope of real normed spaces by means of the f-statistical convergence for f a modulus function. Here, we go several steps further and extend the f-statistical convergence to the scope of uniform spaces, obtaining particular cases of f-statistical convergence on pseudometric spaces and topological modules.Öğe General methods of convergence and summability(Springer, 2021) Javier Garcia-Pacheco, Francisco; Kama, Ramazan; del Carmen Listan-Garcia, MariaThis paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of N and study the space of convergence associated with the filter. We notice that c(X) is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then l(infinity)(X) is a space of convergence associated with any free ultrafilter of N; and that if X is not complete, then l(infinity)(X) is never the space of convergence associated with any free filter of N. Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that l(infinity)(X) is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then c(X) is a space of convergence through a certain class of such operators; and that if X is not complete, then c(X) is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set HB(lim):={T is an element of B(l(infinity)(X),X):T|(c(X))=lim and parallel to T parallel to=1} and prove that HB(lim) is a face of B-LX(0) if X has the Bade property, where L-X(0):={T is an element of B(l(infinity)(X),X):c(0)(X)subset of ker(T)}. Finally, we study the multipliers associated with series for the above methods of convergence.Öğe Multiplier Sequence Spaces Defined by Statistical Summability and Orlicz-Pettis Theorem(Taylor & Francis Inc, 2021) Kama, Ramazan; Altay, BilalIn this paper, we introduce some new multiplier spaces related to a series Sigma(i)x(i) in a normed space X through f-statistical summability and give some characterizations of completeness and barrelledness of the spaces through weakly unconditionally Cauchy series in X and X*, respectively. We also obtain a new version of the Orlicz-Pettis theorem within the frame of the f-statistical convergence.Öğe On Cesàro summability of vector valued multiplier spaces and operator valued series(2018) Altay, Bilal; Kama, RamazanIn this paper, we introduce and study vector valued multiplier spaces with the help of the sequence of continuous linear operators between normed spaces and Cesàro convergence. Also, we obtain a new version of the Orlicz–Pettis Theorem by means of Cesàro summability.Öğe On f-strongly Cesaro and f-statistical derivable functions(Amer Inst Mathematical Sciences-Aims, 2022) Altay, Bilal; Garcia-Pacheco, Francisco Javier; Kama, RamazanIn this manuscript, we introduce the following novel concepts for real functions related to f-convergence and f-statistical convergence: f-statistical continuity, f-statistical derivative, and f-strongly Cesaro derivative. In the first subsection of original results, the f-statistical continuity is related to continuity. In the second subsection, the f-statistical derivative is related to the derivative. In the third and final subsection of results, the f-strongly Cesaro derivative is related to the strongly Cesaro derivative and to the f-statistical derivative. Under suitable conditions of the modulus f, several characterizations involving the previous concepts have been obtained.Öğe On Modulus Statistical Convergence in Partial Metric Spaces(Mdpi, 2024) Garcia-Pacheco, Francisco Javier; Kama, RamazanModulus statistical convergence has been studied in very different general settings such as topological spaces and uniform spaces. In this manuscript, modulus statistical convergence is defined and studied in partial metric spaces.Öğe On Some Vector Valued Multiplier Spaces with Statistical Cesaro Summability(Univ Nis, Fac Sci Math, 2019) Kama, RamazanIn the present paper we define and study some vector valued spaces within the frame of the Statistical Cesaro convergence and the Statistically Cesaro summability in normed spaces. Also, we characterize the continuous, compact and sequential continuous summing operators on Statistical Cesaro multiplier sequence spaces.Öğe On Zweier convergent vector valued multiplier spaces(2019) Kama, RamazanIn this paper, we introduce the Zweier convergent vector valued multiplier spaces ??????(??? ????????)and ???????? (? ???????? ?? ). We study some topological and algebraic properties on these spaces.Furthermore, we study some inclusion relations concerning these spaces.Öğe Spaces of vector sequences defined by the f-statistical convergence and some characterizations of normed spaces(Springer-Verlag Italia Srl, 2020) Kama, RamazanIn this paper, we introduce the spaces of vector valued sequences defined by f-statistical convergence and f-statistical summability. We give some topological properties of these spaces. Also, we characterize the Schur property, the Grothendieck property and reflexivity of a normed space in terms of f-statistical convergence.Öğe Vector-Valued Spaces of Multiplier Statistically Convergent Series and Uniform Convergence(Springer Basel Ag, 2022) Garcia-Pacheco, Francisco Javier; Kama, Ramazan; Murillo-Arcila, MarinaIn this paper, we introduce the spaces of vector-valued sequences containing multiplier (weakly) statistically convergent series. The completeness of such spaces is studied as well as some relations between unconditionally convergent and weakly unconditionally Cauchy series of these spaces. We also obtain generalizations of some results regarding uniform convergence of unconditionally convergent series through the concept of statistical convergence. Finally, we provide a version of the Orlicz- Pettis theorem for A-multiplier convergent operator series by means of the statistical convergence.Öğe Weakly unconditionally Cauchy series and Fibonacci sequence spaces(Springer International Publishing, 2017) Kama, Ramazan; Altay, BilalWe study new sequence spaces associated to sequences in normed spaces and the band matrix F̂ defined by the Fibonacci sequence. We give some characterizations of continuous linear operators and weakly unconditionally Cauchy series by means of completeness of the new sequence spaces. Also, we characterize the barreledness of a normed space via weakly∗ unconditionally Cauchy series in X*.
