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An integer-forcing linear receiver has significantly better performance than conditional receivers for slow-fading channels because it directly recovers an integer linear combination of signals instead of decoding all signals. The performance in terms of achievable rate, outage probability, and error rate can be improved with a unitary matrix precoder imposed at each channel realization. In this paper, a new special unitary matrix precoding approach is proposed to reduce the computational complexity. Different from the parameterization technique with many parameters, the new method constructs a unitary Vandermonde matrix with only a single parameter. The optimal Vandermonde matrix is determined on the basis of the shortest vector of a lattice generated by the precoding matrix in which the single parameter is searched. Therefore, its complexity is reduced to a polynomial time, whereas the traditional unitary precoder has exponential complexity. Simulation results show that the proposed scheme can achieve the performance similar to the benchmark schemes but with much lower complexity. The scheme offers a good trade-off between performance and complexity. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the best approximation theory in Riesz space, which is not constructed by the norm, but only rely on the order structure. Based on the order structure, we propose the concept of the order best approximation in Riesz space and discuss some problems related to the order best approximation, including some sufficient and necessary conditions for satisfying the order best approximation set. Finally, we consider the order best approximation projection and its related properties. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0 < β 0 < β 1 < β 2 < ⋯ < β m < 1 . The problem is first approximated by the L 1 difference method on the temporal direction, and then, the Galerkin–Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2 − β m in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results. [ABSTRACT FROM AUTHOR]

We present theoretical studies for the third-order elastic constants of Mg, Be, Ti, Zn, Zr, and Cd with a hexagonal-close-packed (HCP) structure. The method of homogeneous deformation combined with first-principles total-energy calculations is employed. The deformation gradient F i j is applied to the crystal lattice vectors r i , and the elastic strain energy can be obtained from the first-principles calculation. The second- and third-order elastic constants are extracted by a polynomial fit to the calculated energy-strain results. In order to assure the accuracy of our method, we calculated the complete set of the equilibrium lattice parameters and second-order elastic constants for Mg, Be, Ti, Zn, Zr, and Cd, and our results provide better agreement with the previous calculated and experimental values. Besides, we have calculated the pressure derivatives of SOECs related to third-order elastic constants, and high-pressure effects on elastic anisotropy, ductile-to-brittle criterion, and Vickers hardness are also investigated. The results show that the hardness model H v = 1.877 k 2 G 0.585 is more appropriate than H v = 2 k 2 G 0.585 − 3 for HCP metals under high pressure. [ABSTRACT FROM AUTHOR]

In this article, we introduce and study a new class of operators defined on a Cartesian product of ideal spaces of measurable functions. We use the general approach of the theory of vector lattices. We say that an operator T : E × F ⟶ W defined on a Cartesian product of vector lattices E and F and taking values in a vector lattice W is orthogonally biadditive if all partial operators T y : E ⟶ W and T x : F ⟶ W are orthogonally additive. In the first part of the article, we prove that, under some mild conditions, a vector space of all regular orthogonally biadditive operators O B A r E , F ; W is a Dedekind complete vector lattice. We show that the set of all horizontally-to-order continuous regular orthogonally biadditive operators is a projection band in O B A r E , F ; W . In the last section of the paper, we investigate orthogonally biadditive operators on a Cartesian product of ideal spaces of measurable functions. We show that an integral Uryson operator which depends on two functional variables is orthogonally biadditive and obtain a criterion of the regularity of an orthogonally biadditive Uryson operator. [ABSTRACT FROM AUTHOR]

The article introduces the scale‐deviating operator. The scale‐deviating differential operator comprises the parameters to designate the operator order and the parameters to define the dimension of space. The operator order depends on the characteristic length κ. There are two types of linear scale‐deviating operators. For the distances r, which are much less than κ, the scale‐deviating operator of the first type A reduces to the common operators. For the distances, which exceed the length κ, this operator reduces to the fractional Riesz operator. The second type of the scale‐deviating operator B behaves oppositely. For the distances r, which are much higher than κ, the scale‐deviating operator of the second type reduces to the common operators. Finally, for the distances, which below the length κ, this operator lessens to the fractional Riesz operator. These linear, isotropic operators possess the order, less than two. The solutions of new scale‐deviating equations and the shell theorem for these operators are provided closed form. [ABSTRACT FROM AUTHOR]

In this work, we investigate an effective linearized spectral collocation method for two‐dimensional (2D) Riesz space fractional nonlinear reaction–diffusion equations with homogeneous boundary conditions. The proposed method is based on the Jacobi–Gauss–Lobatto spectral collocation method for spatial discretization and the finite difference method for temporal discretization. The full implementation of the method is demonstrated in detail. The stability of the numerical scheme is rigorously discussed and the errors with benchmark solutions that show second‐order convergence in time and spectral convergence in space are numerically analyzed. Finally, numerical simulations for 2D Riesz space fractional Allen–Cahn and FitzHugh–Nagumo models are carried out to illustrate the effectiveness of the developed method and its ability for long‐time simulations. [ABSTRACT FROM AUTHOR]

In this article, the Riesz space fractional advection‐dispersion equations with delay in two‐dimensional (RFADED in 2D) are considered. The Riesz space fractional derivative is approximated with the aid of backward differential formulas method of second order and shifted Grünwald difference operators. We develop the Crank‐Nicolson scheme using the finite difference method for the RFADED in 2D and show that it is conditionally stable and convergent with the accuracy order O(κ2+h2+k2). Finally, some numerical examples are constructed to demonstrate the efficacy and usefulness of the numerical method. [ABSTRACT FROM AUTHOR]

We prove that if E is a Dedekind complete atomless Riesz space and X is a Banach space, then the sum of two laterally continuous orthogonally additive operators from E to X, one of which is strictly narrow and the other one is hereditarily strictly narrow with finite variation (in particular, has finite rank), is strictly narrow. Similar results were previously obtained for narrow operators by different authors; however, no theorem of the kind was known for strictly narrow operators. [ABSTRACT FROM AUTHOR]

Let L=-Δ+V be a Schrödinger operator, where Δ is the Laplacian on Rd and the nonnegative potential V belongs to the reverse Hölder class RHq for q≥d. The Riesz transform associated with the operator L=-Δ+V is denoted by R=∇(-Δ+V)-1/2 and the dual Riesz transform is denoted by R⁎=(-Δ+V)-1/2∇. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RHq for q≥d. Then we will establish the mapping properties of the operator R and its adjoint R⁎ on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators [b,R] and [b,R⁎] are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger than Ap, BMO(Rd), and Lp,κ(w) corresponding to the classical Riesz transforms (V≡0). [ABSTRACT FROM AUTHOR]

A sequence {φn} in a Hilbert space H with inner product <·,·> is called a generalized Riesz system if there exist an ONB e={en} in H and a densely defined closed operator T in H with densely defined inverse such that {en}⊂D(T)∩D((T-1)⁎) and Ten=φn, n=0,1,⋯, and (e,T) is called a constructing pair for {φn} and T is called a constructing operator for {φn}. The main purpose of this paper is to investigate under what conditions the ordered set Cφ of all constructing operators for a generalized Riesz system {φn} has maximal elements, minimal elements, the largest element, and the smallest element in order to find constructing operators fitting to each of the physical applications. [ABSTRACT FROM AUTHOR]

The inversion of Riesz potentials for Dunkl transform when G=Z2d is given by using the generalized wavelet transforms. It is also proved that the Riesz potentials Iακ are automorphisms on the Semyanistyi-Lizorkin spaces. [ABSTRACT FROM AUTHOR]

We construct some new Riesz bases and consider the stability of them. The investigation is based on the stability of Riesz bases of cosines and sines in the Hilbert space L2[0,π]. [ABSTRACT FROM AUTHOR]

There are many industrial and biological reaction diffusion systems which involve the time-varying features where certain parameters of the system change during the process. A part of the transport-reaction phenomena is often modelled as an abstract nonautonomous equation generated by a (generalized) Riesz-spectral operator on a Hilbert space. The basic problems related to the equations are existence of solutions of the equations and how to control dynamical behaviour of the equations. In contrast to the autonomous control problems, theory of controllability and observability for the nonautonomous systems is less well established. In this paper, we consider some relevant aspects regarding the controllability and observability for the nonautonomous Riesz-spectral systems including the Sturm-Liouville systems using a C0-quasi-semigroup approach. Three examples are provided. The first is related to sufficient conditions for the existence of solutions and the others are to confirm the approximate controllability and observability of the nonautonomous Riesz-spectral systems and Sturm-Liouville systems, respectively. [ABSTRACT FROM AUTHOR]

A new concept of statistically e-uniform Cauchy sequences is introduced to study statistical order convergence, statistically relatively uniform convergence, and norm statistical convergence in Riesz spaces. We prove that, for statistically e-uniform Cauchy sequences, these three kinds of convergence for sequences coincide. Moreover, we show that the statistical order convergence and the statistically relatively uniform convergence need not be equivalent. Finally, we prove that, for monotone sequences in Banach lattices, the norm statistical convergence coincides with the weak statistical convergence. [ABSTRACT FROM AUTHOR]

Let X be a Banach function space over the unit circle T and let H[X] be the abstract Hardy space built upon X. If the Riesz projection P is bounded on X and a∈L∞, then the Toeplitz operator Taf=P(af) is bounded on H[X]. We extend well-known results by Brown and Halmos for X=L2 and show that, under certain assumptions on the space X, the Toeplitz operator Ta is bounded (resp., compact) if and only if a∈L∞ (resp., a=0). Moreover, aL∞≤TaB(H[X])≤PB(X)aL∞. These results are specified to the cases of abstract Hardy spaces built upon Lebesgue spaces with Muckenhoupt weights and Nakano spaces with radial oscillating weights. [ABSTRACT FROM AUTHOR]

Let Ω and I denote a compact metrizable space with card(Ω)≥2 and the unit interval, respectively. We prove Milutin and Cantor-Bernstein type theorems for the spaces M(Ω) of Radon measures on compact Hausdorff spaces Ω. In particular, we obtain the following results: (1) for every infinite closed subset K of βN the spaces M(K), M(βN), and M(Ω2ℵ0) are order-isometric; (2) for every discrete space Γ with m≔card(Γ)>ℵ0 the spaces M(βΓ) and M(I2m) are order-isometric, whereas there is no linear homeomorphic injection from C(βT) into C(I2m). [ABSTRACT FROM AUTHOR]

We depart from the well-known one-dimensional Fisher’s equation from population dynamics and consider an extension of this model using Riesz fractional derivatives in space. Positive and bounded initial-boundary data are imposed on a closed and bounded domain, and a fully discrete form of this fractional initial-boundary-value problem is provided next using fractional centered differences. The fully discrete population model is implicit and linear, so a convenient vector representation is readily derived. Under suitable conditions, the matrix representing the implicit problem is an inverse-positive matrix. Using this fact, we establish that the discrete population model is capable of preserving the positivity and the boundedness of the discrete initial-boundary conditions. Moreover, the computational solubility of the discrete model is tackled in the closing remarks. [ABSTRACT FROM AUTHOR]

We characterize the weighted local Hardy spaces hρ1(ω) related to the critical radius function ρ and weights ω∈A1ρ,∞(Rn) by localized Riesz transforms R^j; in addition, we give a characterization of weighted Hardy spaces HL1(ω) via Riesz transforms associated with Schrödinger operator L, where L=-Δ+V is a Schrödinger operator on Rn (n≥3) and V is a nonnegative function satisfying the reverse Hölder inequality. [ABSTRACT FROM AUTHOR]

We derive and analyze second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations (RSDO-ADE) in one-dimensional (1D) and two-dimensional (2D) cases, respectively. Firstly, we discretize the Riesz space distributed-order advection-dispersion equations into multiterm Riesz space fractional advection-dispersion equations (MT-RSDO-ADE) by using the midpoint quadrature rule. Secondly, we propose a second-order accurate implicit numerical method for the MT-RSDO-ADE. Thirdly, stability and convergence are discussed. We investigate the numerical solution and analysis of the RSDO-ADE in 1D case. Then we discuss the RSDO-ADE in 2D case. For 2D case, we propose a new second-order accurate implicit alternating direction method, and the stability and convergence of this method are proved. Finally, numerical results are presented to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

G-frames and g-Riesz frames as generalized frames in Hilbert spaces have been studied by many authors in recent years. The super Hilbert space has a certain advantage compared with the Hilbert space in the field of studying quantum mechanics. In this paper, for super Hilbert space H⊕K, the definitions of a g-Riesz frame and minimal g-complete are put forward; also a characterization of g-Riesz frames is obtained. In particular, we generalize them to general super Hilbert space L1⊕L2⊕⋯⊕Ln. Finally, a conclusion of the stability of a g-Riesz frame for the super Hilbert space is given. [ABSTRACT FROM AUTHOR]

A scale of spaces exists connecting the class of functions of bounded kth p-variation in the sense of Riesz-Merentes with the Sobolev space of functions with p-integrable kth derivative. This scale is generated by the generalized functionals of Merentes type. We prove some limiting relations for these functionals as well as sharp estimates in terms of the fractional modulus of smoothness of order k-1/p. [ABSTRACT FROM AUTHOR]

In this note, we carry out investigations related to the mixed impact of ordering and topological structure of a locally convex solid Riesz space (X,τ) and a scalar valued sequence space λ, on the vector valued sequence space λ(X) which is formed and topologized with the help of λ and X, and vice versa. Besides,we also characterize o-matrix transformations from c(X), ℓ∞(X) to themselves, cs(X) to c(X) and derive necessary conditions for a matrix of linear operators to transform ℓ1(X) into a simple ordered vector valued sequence space Λ(X).

We obtain an atomic decomposition of weighted Lorentz spaces for a class of weights satisfying the 2 condition. Consequently, we study operators such as the multiplication and composition operators and also provide Hölder's-type and duality-Riesz type inequalities on these weighted Lorentz spaces. [ABSTRACT FROM AUTHOR]

We consider generalized Orlicz-Morrey spaces MΦ, φ(ℝ²) including their weak versions WMMΦ, φ(ℝ²)). In these spaces we prove the boundedness of the Riesz potential from MΦ, φ1(ℝ²) to Mψφ2(ℝ²) and from MΦ, φ(ℝ²) to WMχ, φ(ℝ²). As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness aregiven either in terms of Zygmund-type integral inequalities on (φ1, &#9662), which do not assume any assumption on monotonicity of φ1(x, r), φ2 (x, r) in r. [ABSTRACT FROM AUTHOR]

We give a molecular characterization of the Hardy space associated with twisted convolution. As an application, we prove the boundedness of the local Riesz transform on the Hardy space. [ABSTRACT FROM AUTHOR]

We deal with the stability of the exponential Cauchy functional equation F(x + y) = F(x)F(y) in the class of functions F:G → L mapping a group (G, +) into a Riesz algebra L. The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem. [ABSTRACT FROM AUTHOR]

We firstly develop the high-order numerical algorithms for the left and right Riemann-Liouville derivatives. Using these derived schemes, we can get high-order algorithms for the Riesz fractional derivative. Based on the approximate algorithm, we construct the numerical scheme for the space Riesz fractional diffusion equation, where a fourth-order scheme is proposed for the spacial Riesz derivative, and where a compact difference scheme is applied to approximating the first-order time derivative. It is shown that the difference scheme is unconditionally stable and convergent. Finally, numerical examples are provided which are in line with the theoretical analysis. [ABSTRACT FROM AUTHOR]

We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra). The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equation F(x + y) + F(x) + F(y) ≠ 0 ⇒ F(x + y) = F(x) + F(y) in Riesz spaces, the Cauchy equation with squares F(x + y)² = (F(x) + F(y))² in f-algebras, and the quadratic functional equation F(x + y) + F(x - y) = 2F(x) + 2F(y) in Riesz spaces. [ABSTRACT FROM AUTHOR]

We propose two new compact difference schemes for numerical approximation of the Riemann-Liouville and Riesz derivatives, respectively. It is shown that these formulas have fourth-order convergence order by means of the Fourier transform method. Finally, some numerical examples are implemented to testify the efficiency of the numerical schemes and confirm the convergence orders. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define the notions of ideal convergence, I-bounded for double sequences in setting of locally solid Riesz spaces and study some results related to these notions. We also define the notion of I*-convergence for double sequences in locally solid Riesz spaces and establish its relationship with ideal convergence. [ABSTRACT FROM AUTHOR]

The general form of continuous linear functionals on an Orlicz sequence space 1ϕ (non-separable and non-locally convex in general) is obtained. It is proved that the space hϕ is an M-ideal in 1ϕ.

Our aim in this paper is to deal with the Sobolev embeddings for generalized Riesz potentials of functions inMorrey spaces L(1,φ)(G) over nondoubling measure spaces. [ABSTRACT FROM AUTHOR]

Given the g-orthonormal basis for Hilbert space H, we characterize the g-frames, normalized tight g-frames, and g-Riesz bases in terms of the g-preframe operators. Then we consider the transformations of g-frames, normalized tight g-frames, and g-Riesz bases, which are induced by operators and characterize them in terms of the operators. Finally, we discuss the sums and g-dual frames of g-frames by applying the results of characterizations. [ABSTRACT FROM AUTHOR]

In 2002 Bennett et al. started the investigation to which extent sequence spaces are determined by the sequences of 0s and 1s they that contain. In this relation they defined three types of Hahn properties for sequence spaces: the Hahn property, separable Hahn property, and matrix Hahn property. In general all these three properties are pairwise distinct. If a sequence space E is solid and ({0, 1}N n E)β = Eβ = l¹ then the two last properties coincide. We will show that even on these additional assumptions the separable ahn property and the Hahn property still do not coincide. However if we assume E to be the bounded summability domain of a regular Riesz matrix Rp or a regular nonnegative Hausdorff matrix Hp, then this assumption alone guarantees that E has the Hahn property. For any (infinite) matrix A the Hahn property of its bounded summability domain is related to the strongly nonatomic property of the density dA defined by A. We will find a simple necessary and sufficient condition for the density dA defined by the generalized Riesz matrix Rp,m to be strongly nonatomic. This condition appears also to be sufficient for the bounded summability domain of Rp,m to have the Hahn property. [ABSTRACT FROM AUTHOR]

The boundedness of the various operators on Ḃσ-Morrey spaces is considered in the framework of the Littlewood-Paley decompositions. First, the Littlewood-Paley characterization of Ḃσ-Morrey-Campanato spaces is established. As an application, the boundedness of Riesz potential operators is revisted. Also, a characterization of Ḃσ-Lipschitz spaces is obtained: and, as an application, the boundedness of Riesz potential operators on Ḃσ-Lipschitz spaces is discussed. [ABSTRACT FROM AUTHOR]

Let L = -Ε + V be a Schrödinger operator on Rn (n ≥ 3), where V ≢ 0 is a nonnegative potential belonging to certain reverse Hölder class Bs ≥ for S ≥ n/2. In this paper, we prove the boundedness of commutators RHbf=bRHf - RH(bf) generated by the higher order Riesz transform RH = ∇²(-Ε + V)-1, where b ∊ BMOθ(p), which is larger than the space BMO(Rn). Moreover, we prove that RHb is bounded from the Hardy space H¹L(Rn) into weak L¹weak(Rn). [ABSTRACT FROM AUTHOR]

We present the space of functions of bounded kφ-variation in the sense of Riesz-Korenblum, denoted by kBVφ[a,b], which is a combination of the notions of bounded φ-variation in the sense of Riesz and bounded φ-variation in the sense of Korenblum. Moreover, we prove that the space generated by this class of functions is a Banach space with a given norm and we prove that the uniformly bounded composition operator satisfies Matkowski's weak condition. [ABSTRACT FROM AUTHOR]

Assume that G is a stratified Lie group and Q is the homogeneous dimension of G. Let -△ be the sub-Laplacian on G and W ≠ 0 a nonnegative potential belonging to certain reverse Hölder class Bs for s ≥ Q√2. Let L = -△ + W be a Schrödinger operator on the stratified Lie group G. In this paper, we prove the boundedness of some integral operators related to L, such as L-1▽², L-1W, and L-1(△) on the space BMOL(G). [ABSTRACT FROM AUTHOR]

Let Σ be a σ-algebra of subsets of a nonempty set ω. Let B(Σ) be the complex vector lattice of bounded Σ-measurable complexvalued functions on ω and let(Σ) be the Banach space of all bounded countably additive complex-valued measures on ω. We study locally solid topologies on B(Σ). In particular, it is shown that the Mackey topology τ(B(Σ), ca(Σ)) is the finest locally convexsolid σ-Lebesgue topology on B(Σ). [ABSTRACT FROM AUTHOR]

We introduce the modular continuous g-Riesz basis to improve one existing result for continuous g-Riesz basis in Hilbert C*- modules, and then we study the equivalency relations between continuous g-frames in Hilbert C*-modules, and, in particular, we obtain two necessary and sufficient conditions under which two continuous g-frames are similar. Finally, we generalize a stability result for alternate duals of g-frames in Hilbert spaces to alternate duals of continuous g-frames in Hilbert C*-modules. [ABSTRACT FROM AUTHOR]

A method to measure the temperature of a metal plate is presented using a Riesz transform method and the monogenic signal to extract the optical phase distribution from a fringe pattern from which one can get the unknown temperature. The performance of this method is evaluated by the RFSIM metric obtained from the 2nd-order Riesz transform. A phase distribution with a good accuracy is provided. [ABSTRACT FROM AUTHOR]

We study almost everywhere convergence for Riesz means related to Schrödinger operator with constant magnetic fields. Through researching the weighted norm estimates for the maximal operator with power-weight functions, we obtain the desired result, which is similar to the work given by Anthony Carbery, Jose L. Rubio de Francia, and Luis Vega. [ABSTRACT FROM AUTHOR]

We define the notions of double statistically convergent and double lacunary statistically convergent sequences in locally solid Riesz space and establish some inclusion relations between them. We also prove an extension of a decomposition theorem in this setup. Further, we introduce the concepts of double θ-summable and double statistically lacunary summable in locally solid Riesz space and establish a relationship between these notions. [ABSTRACT FROM AUTHOR]

Let L = -Δ + V be a Schrödinger operator, where Δ is the laplacian on ...Rn and the nonnegative potential V belongs to the reverse Hölder class BV s1 for some sV 1V ≥ (n/2). Assume that ω ∊ AV 1V (...Rn). Denote by H¹V LV (ω) the weighted Hardy space related toV the Schrödinger operator L = -Δ+V. Let RbV = [b,R] be the commutator generated by a function b ∊ BMOθ V(...Rn) and the Riesz V transform R = ∇(-Δ+V)-(1/2). Firstly, we show that the operatorRis bounded fromL¹(ω) into L¹Vweak(ω). Secondly, we obtain theVendpoint estimates for the commutator [b,R]. Namely, it is bounded from the weighted Hardy space H¹VLV(ω) into L¹Vweak(ω). [ABSTRACT FROM AUTHOR]

We give new characterizations of Riesz-type frames, on equivalent conditions for a continuous frame to be a Riesz-type frame and on equivalency relations between Riesz-type frames and continuous frames. We characterize also the Riesz-type frames by using a bounded linear operator L. Finally, we study the stability of alternate duals of continuous frames and we prove that if two continuous frames are close to each other, then we can find alternate duals of them which are close to each other. [ABSTRACT FROM AUTHOR]

We introduce and study the concept of (p, k)-variation (1 < p < ∞, k ∈ N) of a real function on a compact interval. In particular, we prove that a function u : [a, b] → R has bounded (p, k)-variation if and only if u(k-1) is absolutely continuous on [a, b] and u(k) belongs to Lp[a, b]. Moreover, an explicit connection between the (p, k)-variation of u and the Lp-norm of u(k) is given which is parallel to the classical Riesz formula characterizing functions in the spaces RVp[a, b] and Ap[a, b]. This may also be considered as an alternative characterization of the one variable Sobolev space Wpk [a, b]. [ABSTRACT FROM AUTHOR]

Recently, the notion of statistical convergence is studied in a locally solid Riesz space by Albayrak and Pehlivan (2012). In this paper, we define and study statistical τ-convergence, statistical τ-Cauchy and S* (τ)-convergence of double sequences in a locally solid Riesz space. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce new spaces ˆf and ˆf0 that consist of all sequences whose Riesz transforms of order one are in the spaces f and f0, respectively. We also show that ˆf and ˆ f0 are linearly isomorphic to the spaces f and f0, respectively. The ß- and Γ-duals of the spaces ˆf and ˆ f0 are computed. Furthermore, the classes (ˆ f : µ) and (µ : ˆf) of infinite matrices are characterized for any given sequence space µ and determine the necessary and sufficient conditions on a matrix A to satisfy BR - core(Ax) ⊆ K - core(x), BR - core(Ax) ⊆ st - core(x) for all x ∈ℓ∞. [ABSTRACT FROM AUTHOR]

Full description of Riesz basis property for eigenfunctions of boundary value problems for first order differential equations with involutions is given. [ABSTRACT FROM AUTHOR]