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New concepts of $f_{\lambda,\mu }$ -statistical convergence for double sequences of order α̃ and strong $f_{\lambda,\mu }$ -Cesàro summability for double sequences of order α̃ are introduced for sequences of (complex or real) numbers. Furthermore, we give the relationship between the spaces $w_{\tilde{\alpha },0}^{2} ( f,\lambda,\mu )$ , $w_{\tilde{\alpha }}^{2} ( f,\lambda,\mu ) $ and $w_{\tilde{\alpha},\infty }^{2} ( f,\lambda,\mu )$ . Then we express the properties of strong $f_{\lambda,\mu }$ -Cesàro summability of order β̃ which is related to strong $f_{\lambda,\mu }$ -Cesàro summability of order α̃. Also, some relations between $f_{\lambda,\mu }$ -statistical convergence of order α̃ and strong $f_{\lambda,\mu }$ -Cesàro summability of order α̃ are given. [ABSTRACT FROM AUTHOR]

In Hardy (Proc Camb Philos 19:86-95, 1916), Hardy defined the normed spaces $$\mathcal {C}_{r}$$ and $$\mathcal {C}_{r0}$$ of all regularly convergent and regularly null double sequences and made convergence factor calculations (i.e. some beta-dual calculations). In this paper, we extend these spaces to the paranormed spaces $$\mathcal {C}_{r}(t)$$ and $$\mathcal {C}_{r0}(t)$$ . Also, we define the paranormed spaces $$\mathcal {C}_{tr}(t)$$ and $$\mathcal {C}_{tr0}(t)$$ of all totally regularly convergent and totally regularly null double sequences. We examine some topological properties of these spaces and determine their alpha-, beta- and gamma-duals. [ABSTRACT FROM AUTHOR]

The space was introduced and studied by Mursaleen []. In the present article we study the double sequence spaces , and with the help of the space and an Orlicz function M, where is the double sequence of strictly positive real numbers. We study some of its properties and prove some inclusion relations related to these new spaces. [ABSTRACT FROM AUTHOR]

A trigonometric series strongly bounded at two points and with coefficients forming a log-quasidecreasing sequence is necessarily the Fourier series of a function belonging to all $${L^{p}}$$ spaces, $${1\leq p < \infty}$$ . We obtain new results on strong convergence of Fourier series for functions of generalized bounded variation. [ABSTRACT FROM AUTHOR]

In this study, I introduce some new double sequence spaces $B(\mathcal {M}_{u})$ , $B(\mathcal{C}_{p})$ , $B(\mathcal{C}_{bp})$ , $B(\mathcal {C}_{r})$ and $B(\mathcal{L}_{q})$ as the domain of four-dimensional generalized difference matrix $B(r,s,t,u)$ in the spaces $\mathcal {M}_{u}$ , $\mathcal{C}_{p}$ , $\mathcal{C}_{bp}$ , $\mathcal{C}_{r}$ and $\mathcal{L}_{q}$ , respectively. I show that the double sequence spaces $B(\mathcal{M}_{u})$ , $B(\mathcal{C}_{bp})$ and $B(\mathcal{C}_{r})$ are the Banach spaces under some certain conditions. I give some inclusion relations with some topological properties. Moreover, I determine the α-dual of the spaces $B(M_{u})$ and $B(\mathcal{C}_{bp})$ , the $\beta(\vartheta)$ -duals of the spaces $B(M_{u})$ , $B(\mathcal{C}_{p})$ , $B(\mathcal{C}_{bp})$ , $B(\mathcal{C}_{r})$ and $B(\mathcal{L}_{q})$ , where $\vartheta\in\{p,bp,r\}$ , and the γ-dual of the spaces $B(\mathcal{M}_{u})$ , $B(\mathcal{C}_{bp})$ and $B(\mathcal{L}_{q})$ . Finally, I characterize the classes of four-dimensional matrix mappings defined on the spaces $B(\mathcal{M}_{u})$ , $B(\mathcal{C}_{p})$ , $B(\mathcal{C}_{bp})$ , $B(\mathcal{C}_{r})$ and $B(\mathcal{L}_{q})$ of double sequences. [ABSTRACT FROM AUTHOR]

The volume of the unit ball of the Lebesgue sequence space [ABSTRACT FROM AUTHOR]

In this paper, which deals essentially with various summability concepts and summability techniques and shows how these concepts and techniques lead to a number of approximation results, we have used the new concept of weighted A -summability proposed by Mohiuddine (2016) and introduced the notions of statistically weighted B -summability and weighted B -statistical convergence with respect to the weighted regular method. We then prove a Korovkin type approximation theorem for functions of two variables and also present an example via generalized Meyer-König and Zeller type operator to show that our proposed method is stronger than its classical and statistical versions. Furthermore, the rate of convergence of approximating positive linear operators are estimated by means of the modulus of continuity and some Voronovskaja type results are investigated. Computational and geometrical approaches to illustrate some of our results are also presented. [ABSTRACT FROM AUTHOR]

Let E be a Banach lattice on $${\mathbb {Z}}$$ with order continuous norm. We show that for any function $$f = \{f_j\}_{j \in {\mathbb {Z}}}$$ from the Hardy space $$\mathrm H_{\infty }\left( E \right) $$ such that $$\delta \leqslant \Vert f (z)\Vert _E \leqslant 1$$ for all z from the unit disk $${\mathbb {D}}$$ there exists some solution $$g = \{g_j\}_{j \in {\mathbb {Z}}} \in \mathrm H_{\infty }\left( E' \right) $$ , $$\Vert g\Vert _{\mathrm H_{\infty }\left( E' \right) } \leqslant C_\delta $$ of the Bézout equation $$\sum _j f_j g_j = 1$$ , also known as the vector-valued corona problem with data in $$\mathrm H_{\infty }\left( E \right) $$ . [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the sequence spaceusing a new regular matrix of Fibonacci numbers, where. We investigate its topological properties such as Schauder basis,andduals and its geometric properties like uniformly convex, strictly convex and super-reflexive. We also characterize some matrix classes from thespace to classical sequence spaces. [ABSTRACT FROM PUBLISHER]

Distance measures play a central role in evolving the clustering technique. Due to the rich mathematical background and natural implementation of $l_{p}$ distance measures, researchers were motivated to use them in almost every clustering process. Beside $l_{p}$ distance measures, there exist several distance measures. Sargent introduced a special type of distance measures $m(\phi)$ and $n(\phi)$ which is closely related to $l_{p}$ . In this paper, we generalized the Sargent sequence spaces through introduction of $M(\phi)$ and $N(\phi)$ sequence spaces. Moreover, it is shown that both spaces are BK-spaces, and one is a dual of another. Further, we have clustered the two-moon dataset by using an induced $M(\phi)$ -distance measure (induced by the Sargent sequence space $M(\phi)$ ) in the k-means clustering algorithm. The clustering result established the efficacy of replacing the Euclidean distance measure by the $M(\phi)$ -distance measure in the k-means algorithm. [ABSTRACT FROM AUTHOR]

A mixed-norm version of Minkowski's integral inequality is proved in detail, and combined with the mixed-norm Hölder's inequality to produce new, more general estimates involving symmetric geometric means of mixed norms. Various existing mixed-norm estimates are shown to follow as special cases. Examples show how other estimates can be easily proved using these mixed-norm inequalities. Finally, the effectiveness of this technique is demonstrated by deriving a new inequality, combining features from two previous results. [ABSTRACT FROM AUTHOR]

The main object of this paper is to introduce and study a new concept of f-Wijsman lacunary statistical convergence of sequences of sets, where f is an unbounded modulus. The definition of Wijsman lacunary strong convergence of sequences of sets is extended to a definition of Wijsman lacunary strong convergence with respect to a modulus for sequences of sets and it is shown that, under certain conditions on a modulus f, the concepts of Wijsman lacunary strong convergence with respect to a modulus f and f-Wijsman lacunary statistical convergence are equivalent on bounded sequences. We further characterize those θ for which $\mathit{WS}_{\theta}^{f} = \mathit{WS}^{f}$ , where $\mathit{WS}_{\theta}^{f}$ and $\mathit{WS}^{f}$ denote the sets of all f-Wijsman lacunary statistically convergent sequences and f-Wijsman statistically convergent sequences, respectively. [ABSTRACT FROM AUTHOR]

In the present paper, the difference sequence spaces c s 0 λ ( Δ ) , c s λ ( Δ ) and bs λ ( Δ ) of nonabsolute type are generalized by introducing a generalized Λ difference operator Λ ( Δ ( α ˜ ) ) . Also, their Schauder basis are calculated and α -, β - and γ -duals of these spaces are investigated. Finally, some matrix transformations between these spaces and the basic sequence spaces ℓ p , c and c 0 are characterized, where 1 ≤ p ≤ ∞. [ABSTRACT FROM AUTHOR]

Recently, Kórus [7] studied the $${\Lambda^{2}}$$ -strong convergence of numerical sequences. By using the idea of this paper, we introduce $${[\Lambda^{2}, F, u, p]}$$ -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. [ABSTRACT FROM AUTHOR]

The paper investigates some classes of real number sequences over 2-normed spaces defined by means of Orlicz functions, a bounded sequence of strictly positive real numbers, a multiplier and a normal paranormed sequence space. Relevant properties of such classes have been investigated. Moreover, relationships among different such classes of sequences have also been studied under various parameters and conditions. Finally, the spaces are investigated for some other useful properties. The conclusion section provides many interesting facts for further research. [ABSTRACT FROM AUTHOR]

In this paper, we define new spaces as a generalization of the Fibonacci difference sequence spaces. Also, we establish some inclusion theorems related to these spaces and find the-,-,-duals. Lastly, we characterize some matrix classes on these spaces. [ABSTRACT FROM PUBLISHER]

In this paper we introduce the spaces V ^ λ [ A , M , Δ , p ] o , V ^ λ [ A , M , Δ , p ] and V ^ λ [ A , M , Δ , p ] ∞ generated by infinite matrices defined by Orlicz functions. Also we introduce the concept of S ^ λ [ A , Δ ] − convergence and derive some results between the spaces S ^ λ [ A , Δ ] and V ^ λ [ A , Δ ] . Further, we study some geometrical properties such as order continuity, the Fatou property and the Banach–Saks property of the new space V ^ λ α [ A , Δ , p ] ∞ . Finally, we introduce the notion of almost λ -statistically-[ A, Δ ]-convergence of order α or S ^ λ α [ A , Δ ] − convergence and obtain some inclusion relations between the set S ^ λ α [ A , Δ ] and the space V ^ λ α [ A , Δ , p ] ∞ . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a brief discussion of both the normed space of operator p-summable sequences in a Banach space and the normed space of sequentially p-limited operators. The focus is on proving that the vector space of all operator p-summable sequences in a Banach space is a Banach space itself and that the class of sequentially p-limited operators is a Banach operator ideal with respect to a suitable ideal norm- and to discuss some other properties and multiplication results of related classes of operators. These results are shown to fit into a general discussion of operator [Y,p]-summable sequences and relevant operator ideals. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce a new difference operator $$B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})$$ , defined by $$(B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})x)_k=r_kx_k+s_{k-1}x_{k-1}+t_{k-2}x_{k-2}+u_{k-3}x_{k-3},$$ where $$\tilde{r},\tilde{s},\tilde{t},\tilde{u}$$ are convergent sequences of real numbers satisfying certain conditions and any term with negative subscript is equal to zero. In fact, as the generalizations of most of the difference operators, the operator $$B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})$$ includes the difference operators $$\Delta ^{1}, \Delta ^{2}, \Delta ^{3}, \Delta _\nu ,\Delta _{uv},B(r,s)$$ and B( r, s, t). We determine certain Toeplitz matrices such as Fibonacci and other summable matrices which can be obtained immediately by taking the inverse of the difference operator $$B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})$$ under some limiting conditions. Moreover, their spectral and other basic properties have been studied. [ABSTRACT FROM AUTHOR]

In this paper, some known results on the absolute Riesz summability factors of infinite series and Fourier series have been proved under weaker conditions. Some new and known results are also obtained. [ABSTRACT FROM AUTHOR]

We obtained necessary and sufficient conditions for the equivalence of two general summability methods. Some new and known results are also obtained. [ABSTRACT FROM AUTHOR]

The main purpose of the paper is to generalize the notions of the Köthe-Toeplitz duals and Null duals of sequence spaces by introducing the concepts of αEF-, βEF-, γEF-duals and NEF-duals, where E = ( E ) and F = ( F ) are two partitions of finite subsets of the positive integers. These duals are computed for the classical sequence spaces l and c . The other purpose of the paper is to introduce the sequence spaces . where $$X \in \left\{ {{l_\infty },\;c,\;{c_0}} \right\}$$ . We investigate the topological properties of these spaces, establish some inclusion relations between them, and compute the NEF-(or Null) duals for these spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with problems in the context of the theoretical foundation of adaptive algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of d-dimensional manifolds and investigate some analytical properties of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $$\Psi $$ on domains or manifolds $$\Gamma $$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ , provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity. For this purpose, a theory of almost diagonal matrices on related sequence spaces $$b^\alpha _{p,q}(\nabla )$$ of Besov type is developed. [ABSTRACT FROM AUTHOR]

The packing constant is an important and interesting geometric parameter of Banach spaces. Inspired by the packing constant for Orlicz sequence spaces, the main purpose of this paper is calculating the Kottman constant and the packing constant of the Cesàro-Orlicz sequence spaces $$({\text{ce}}{{\text{s}}_\varphi })$$ defined by an Orlicz function φ equipped with the Luxemburg norm. In order to compute the constants, the paper gives two formulas. On the base of these formulas one can easily obtain the packing constant for the Cesàro sequence space ces and some other sequence spaces. Finally, a new constant $$\widetilde D$$ (X), which seems to be relevant to the packing constant, is given. [ABSTRACT FROM AUTHOR]

It is well-known that the state space isomorphism theorem fails in infinite-dimensional Hilbert spaces: there exist minimal discrete-time systems (with Hilbert space state spaces) which have the same impulse response, but which are not isomorphic. We consider discrete-time systems on locally convex topological vector spaces which are Hausdorff and barrelled and show that in this setting the state space isomorphism theorem does hold. In contrast to earlier work on topological vector spaces, we consider a definition of minimality based on dilations and show how this necessitates certain definitions of controllability and observability. [ABSTRACT FROM AUTHOR]

The essence of the notion of lineability and spaceability is to find linear structures in somewhat chaotic environments. The existing methods, in general, use ad hoc arguments and few general techniques are known. Motivated by the search of general methods, in this paper we formally extend recent results of G. Botelho and V.V. Fávaro on invariant sequence spaces to a more general setting. Our main results show that some subsets of invariant sequence spaces contain, up to the null vector, a closed infinite-dimensional subspace. [ABSTRACT FROM AUTHOR]

In the present paper we introduce $$\mu $$ -statistical convergent double lacunary sequence spaces defined by a sequence of Orlicz function $$\mathcal {M } = (M_{kl})$$ over $$n$$ -normed spaces. We also make an effort to study some topological properties and inclusion relations between these spaces. [ABSTRACT FROM AUTHOR]

In this paper, following a very recent and new approach of Aizpuru et al. (Quaest. Math. 37:525-530, 2014), we further generalize a concept of α-density to that of $f_{\alpha}$-density, where f is an unbounded modulus and $0 < \alpha\leq1$. As a consequence, we obtain a new nonmatrix convergence method, namely f-statistical convergence of order α or $S_{\alpha}^{f}$-convergence, which is intermediate between the ordinary convergence and the statistical convergence of order α. We also introduce a new concept of strong Cesàro summability of order α with respect to a modulus function f, and finally we investigate the relationship between the set $S_{\alpha}^{f}$ of all f-statistically convergent sequences of order α and the set $w_{\alpha}^{f}$ of all strongly Cesàro summable sequences of order α with respect to f. [ABSTRACT FROM AUTHOR]

In the present paper we introduce and study some generalized paranormed sequence spaces defined by Musielak-Orlicz functions as well as by a sequence of modulus functions. We also study some topological properties and prove some inclusion relations between these spaces. [ABSTRACT FROM AUTHOR]

In this paper, we propose to introduce a new Λ 2 -weighted statistical convergence. Based upon this definition, we prove some Korovkin type theorems. We also find the rate of the convergence for this kind of weighted statistical convergence and derive some Voronovskaya type theorems. [ABSTRACT FROM AUTHOR]

In this paper we have introduced a distance on the space of all two dimensional interval vectors and seen that it is complete. Furthermore, we introduce the null, convergent and bounded sequence spaces of two dimensional interval vectors and some theorems on sequence spaces of two dimensional interval vectors. Also some inclusion relations are studied. [ABSTRACT FROM AUTHOR]

In this paper, we generalize some known theorems by using a general class of quasi power increasing sequences, which is a wider class of sequences, instead of an almost increasing sequence. These theorems also include some known and new results. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the matrix classes $(\ell _{1},\ell _{p}(\widehat{F}))$ ( $1\leq p<\infty $), where $\ell _{p}(\widehat{F})$ is some Fibonacci difference sequence spaces. We also obtain estimates for the norms of the bounded linear operators $L_{A}$ defined by these matrix transformations and find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and examine the concepts of I2( u)-convergence of order (α,β) and uniformly strong p-Cesàro summability of order (α,β), of double sequences of complex (or real) numbers, where are real numbers such that . Also, we establish the main results and prove some inclusion relations between uniformly strong p-Cesàro summability and I2( u)-convergence of order (α,β). [ABSTRACT FROM AUTHOR]

In the Valdivia-Vogt structure tables presented in Ortner and Wagner (J Math Anal Appl 404(1):1-10, ) there are two gaps. We fill in these gaps by proving the representations $$\mathcal {D}^{F}\cong s\widehat{\otimes }_{\pi }\mathbb {C}^{(\mathbb {N})}$$ and $$\dot{\mathcal {B}}' \cong s'\widehat{\otimes } c_{0}$$ , where $$\mathcal {D}^{F}$$ is a pre-dual space of the space of distributions of finite order introduced by J. Horváth and the space $$\dot{\mathcal {B}}'$$ is the space of distributions vanishing at infinity introduced by L. Schwartz. [ABSTRACT FROM AUTHOR]

In this article we introduce the sequence spaces $$\mathcal Z ^{I}(f)$$ , $$\mathcal Z ^{I}_{0}(f)$$ and $$\mathcal Z ^{I}_{\infty }(f)$$ for a modulus function $$f$$ and study some of the topological and algebraic properties on these spaces. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new generalized absolute summability method and some new definitions on classes of pairs of sequences. By using these new definitions, we generalize the result of Bor (Math Commun 13:21-25, ) dealing with $$\left| C,\alpha ,\gamma ;\beta \right| _{k}$$ summability factors of infinite series. [ABSTRACT FROM AUTHOR]