Generalized vector-valued paranormed sequence space using modulus function

In this paper we introduce a generalized vector-valued paranormed sequence space N_p(E_k,Δ^m,f,s) using modulus function f, where p=(p_k) is a bounded sequence of positive real numbers such that inf_kp_k>0,(E_k,q_k) is a sequence of seminormed spaces with E_k+1⊆E_k for each k ∈ N and s?0. We have also studied sequence space N_p(E_k,Δ^m,f^r,s), where f^r=f°f°f°,…,f (r-times composition of f with itself) and r∈N={1,2,3,…}. Results regarding completeness, K-space, normality, inclusion relations etc. are derived. Further, a study of multiplier of the set N_p(E_k,f,s) is also made by choosing (E_k,‖·‖_k) as sequence of normed algebras.     

Some topological and geometrical properties of the sequence space e (u,p)

In this paper, we introduce the sequence space e^r(u,p) and investigate its some topological and geometrical properties such as basis, α-, β-, γ- duals and the uniform Opial property.     

On some new paranormed euler sequence spaces and Euler Core

In this paper, the sequence spaces e0^τ（u, p） and ec^τ（u, p） of non-absolute type which are the generalization of the Maddox sequence spaces have been introduced and it is proved that the spaces e0^τ（u,p） and ec^τ（u,p） are linearly isomorphic to spaces co（p） and c（p）, respectively. Furthermore, the α-, β- and γ-duals of the spaces 0^τ（u,p） and ec^τ（u,p） have been computed and their bases have been constructed and some topological properties of these spaces have been investigated. Besides this, the class of matrices （e0^τ）（u, p） ： μ） has been characterized, where μ is one of the sequence spaces l∞, c and co and derives the other characterizations for the special cases of μ. In the last section, Euler Core of a complex-valued sequence has been introduced, and we prove some inclusion theorems related to this new type of core.     

Matrix Transformations of the Maddox Vector-Valued Sequence Spaces

In this paper, we give the matrix characterizations from any normal vector-valued FK-space containing ø(X) into scalar-valued sequence space c(q) and by applying this result, we also obtain necessary and sufficient conditions for infinite matrices mapping the sequence spaces , and F_r(X,p) into the space c(q), where p = (p_k) and q = (q_k) are bounded sequences of positive real numbers and r ≥ 0.AMS Subject Classification (2000): 46A45.     

Matrix Transformations of the Maddox Vector-Valued Sequence Spaces

In this paper, we give the matrix characterizations from any normal vector-valued FK-space containing ø(X) into scalar-valued sequence space c(q) and by applying this result, we also obtain necessary and sufficient conditions for infinite matrices mapping the sequence spaces , and F_r(X, p) into the space c(q), where p = (Pk) and q = (qk) are bounded sequences of positive real numbers and r 0.AMS Subject Classification (2000): 46A45.     

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

On the m order difference sequence space of generalized weighted mean and compact operators

In this article, using generalized weighted mean and difference matrix of order m, we introduce the paranormed sequence space ?(u, v, p; Δ^(m)), which consist of the sequences whose generalized weighted Δ^(m)-difference means are in the linear space ?(p) defined by I.J. Maddox. Also, we determine the basis of this space and compute its α-, β- and γ-duals. Further, we give the characterization of the classes of matrix mappings from ?(u, v, p, Δ^(m)) to ?_∞, c and c₀. Finally, we apply the Hausdorff measure of noncompacness to characterize some classes of compact operators given by matrices on the space ?_p(u, v, Δ^(m))(1 ≤ p < ∞).     

Primarite DeL p(L r), 1<p,r<∞

In this article we show thatL ^p(L ^r) is primary forp andr in ]1,+∞[. If (h _k) _k≧1 denote the Haar basis, we begin with a study of the sequence (h _k ⊗h _i) and, in particular, the space generated by a subsequence of this sequence. In the first part we study the base ofL ^p(L ^r) and in the second part we show that this space is primary.     

Some topological and geometric properties of generalized Euler sequence space

In this paper, we introduce the Euler sequence space e ^r (p) of nonabsolute type and prove that the spaces e ^r (p) and l(p) are linearly isomorphic. Besides this, we compute the α-, β- and γ-duals of the space e ^r (p). The results proved herein are analogous to those in [ALTAY, B.—BASŠAR, F.: On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 26 (2002), 701–715] for the Riesz sequence space r ^q (p). Finally, we define a modular on the Euler sequence space e ^r (p) and consider it equipped with the Luxemburg norm. We give some relationships between the modular and Luxemburg norm on this space and show that the space e ^r (p) has property (H) but it is not rotund (R).     

Another look at Cesàro sequence spaces

We consider the Cesàro sequence space cesp as a closed subspace of the infinite ?p-sum of finite dimensional spaces. We easily obtain many known results, for example, cesp has property (β) of Rolewicz, uniform Opial property, and weak uniform normal structure. We also consider some generalized Cesàro sequence spaces. Finally, we compute the von Neumann-Jordan and James constants of the two-dimensional Cesàro sequence space when 1<p?2.     

The Br property and chromatic numbers of generalized graphs

The concept of property B_r is introduced, generalizing property B due to E. Miller. Bounds are established for the minimum number m_r(p), such that there exists a family F of m_r(p) sets, each consisting of p elements and F not having the property B_r.     

A criterion for regular sequences

LetR be a commutative noetherian ring and ƒ₁, …, ƒ_r ∃ R. In this article we give (cf. the Theorem in §2) a criterion for ƒ₁, …, ƒ_r to be regular sequence for a finitely generated module overR which strengthens and generalises a result in [2]. As an immediate consequence we deduce that if V(g ₁, …,g _r ) ⊆ V(ƒ₁, …, ƒ_r) in SpecR and if ƒ₁, …, ƒ_r is a regular sequence inR, theng ₁, …,g _r is also a regular sequence inR.     

Wavelet bases in the Lebesgue spaces on the field of p-adic numbers

In this paper we prove that p-adic wavelets form an unconditional basis in the space L ^r (? _p ⁿ ) and give the characterization of the space L ^r (? _p ⁿ ) in terms of Fourier coefficients of p-adic wavelets.Moreover, the Greedy bases in the Lebesgue spaces on the field of p-adic numbers are also established.     

Some new spaces of double sequences

In this study, we define the double sequence spaces BS, BS(t), CSp, CSbp, CSr and BV, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are complete paranormed or normed spaces under some certain conditions. We determine the α-duals of the spaces BS, BV, CSbp and the β(?)-duals of the spaces CSbp and CSr of double series. Finally, we give the conditions which characterize the class of four-dimensional matrix mappings defined on the spaces CSbp, CSr and CSp of double series.     

The Ramsey property for C p (X)

For a Tychonoff space X, we denote by C _p (X) the space of real-valued continuous functions with the topology of pointwise convergence. We show that (a) Arhangel℉skii℉s property (α ₂) and the Ramsey property introduced by Nogura and Shakhmatov are equivalent for C _p (X), (b) the Ramsey property and Nyikos’ property (α _3/2) are not equivalent for C _p (X). These results answer questions posed by Shakhmatov. Concerning properties (α _i ) for C _p (X), some results on Scheepers’ conjecture are also given.     

QN-spaces, wQN-spaces and covering properties

The main results of the paper are as follows: covering characterizations of wQN-spaces, covering characterizations of QN-spaces and a theorem saying that Cp(X) has the Arkhangel'ski?ˇ property (α₁) provided that X is a QN-space. The latter statement solves a problem posed by M. Scheepers [M. Scheepers, Cp(X) and Arhangel'ski?ˇ's αi-spaces, Topology Appl. 89 (1998) 265-275] and for Tychonoff spaces was independently proved by M. Sakai [M. Sakai, The sequence selection properties of Cp(X), Preprint, April 25, 2006]. As the most interesting result we consider the equivalence that a normal topological space X is a wQN-space if and only if X has the property S₁(Γshr,Γ). Moreover we show that X is a QN-space if and only if Cp(X) has the property (α₀), and for perfectly normal spaces, if and only if X has the covering property (β₃).     

A kind of multiquadric quasi-interpolation operator satisfying any degree polynomial reproduction property to scattered data

In this paper, by virtue of using the linear combinations of the shifts of f(x) to approximate the derivatives of f(x) and Waldron’s superposition idea (2009), we modify a multiquadric quasi-interpolation with the property of linear reproducing to scattered data on one-dimensional space, such that a kind of quasi-interpolation operator L_r+1f has the property of r+1(r∈Z,r≥0) degree polynomial reproducing and converges up to a rate of r+2. There is no demand for the derivatives of f in the proposed quasi-interpolation L_r+1f, so it does not increase the orders of smoothness of f. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operators with that of Wu-Schaback’s quasi-interpolation scheme and Feng-Li’s quasi-interpolation scheme.     

Congruences of multipartition functions modulo powers of primes

Let p _r (n) denote the number of r-component multipartitions of n, and let S _γ,λ be the space spanned by η(24z) ^γ ?(24z), where η(z) is the Dedekind’s eta function and ?(z) is a holomorphic modular form in \(M_{\lambda}(\mathrm{SL}_{2}(\mathbb{Z}))\) . In this paper, we show that the generating function of \(p_{r}(\frac{m^{k} n +r}{24})\) with respect to n is congruent to a function in the space S _γ,λ modulo m ^k . As special cases, this relation leads to many well known congruences including the Ramanujan congruences of p(n) modulo 5,7,11 and Gandhi’s congruences of p ₂(n) modulo 5 and p ₈(n) modulo 11. Furthermore, using the invariance property of S _γ,λ under the Hecke operator \(T_{\ell^{2}}\) , we obtain two classes of congruences pertaining to the m ^k -adic property of p _r (n).     

A note on the sequence spacel (p v )

In this paper, the linear isometry of the sequence space l(p_v) into itself is specified as the automorphism of l(p_v) onto itself, when (p_v) satisfies the conditions, (i) 0 < p_v? 1, (ii) 1 +d ? p_v ? p < ∞,q < q_v < 1+d/d,d > o When (p_v) satisfies condition (ii),l (p_v) andl (q_v) are proved to be perfect spaces in the sense of Kothe and Toeplitz. A similar result connecting linear isometry and automorphism has been noted in the case of a non-normable complete linear metric space whose conjugate space is also determined.