The full lattice convergence on a locally solid Riesz space is an abstraction of the topological, order, and relatively uniform convergences. We investigate four modifications of a full convergence $\mathbb{c}$ on a Riesz space. The first one produces a sequential convergence $\mathbb{sc}$. The second makes an absolute $\mathbb{c}$-convergence and generalizes the absolute weak convergence. The third modification makes an unbounded $\mathbb{c}$-convergence and generalizes various unbounded convergences recently studied in the literature. The last one is applicable whenever $\mathbb{c}$ is a full convergence on a commutative $l$-algebra and produces the multiplicative modification $\mathbb{mc}$ of $\mathbb{c}$. We study general properties of full lattice convergence with emphasis on universally complete Riesz spaces and on Archimedean $f$-algebras. The technique and results in this paper unify and extend those which were developed and obtained in recent literature on unbounded convergences.
Some types of statistical convergence such as statistical order and deferred statistical convergences have been studied and investigated in Riesz spaces, recently. In this paper, we introduce the concept of deferred statistical convergence in Riesz spaces with order convergence. Moreover, we give some relations between deferred statistical order convergence and other kinds of statistical convergences.
The statistical convergence is handled for sequences with the natural density, in general. In a recent paper, the statistical convergence for nets in Riesz spaces has been studied and investigated by developing topology-free techniques in Riesz spaces. In this paper, we introduce the statistically topological convergence for nets on locally solid Riesz spaces with solid topologies. Moreover, we introduce the statistical continuity on locally solid Riesz spaces.
The statistically unbounded $p$-convergence is an abstraction of the statistical order, unbounded order, and $p$-convergences. We investigate the concept of the statistically unbounded convergence on lattice-normed Riesz spaces with respect to statistical p-decreasing sequences. Also, we get some relations between this concept and the other kinds of statistical convergences on Riesz spaces.
The statistical convergence is defined for sequences with the asymptotic density on the natural numbers, in general. In this paper, we introduce the statistical convergence for nets in Riesz spaces by using the finite additive measures on directed sets. Moreover, we give some relations among the statistical convergence and the lattice properties such as the order convergence and lattice operators.
In this paper, we investigate the convergence of the Bochner-Riesz means on some Sobolev type spaces including $L^p$-Sobolev spaces $(p\geq 1)$ and $H^q$-Sobolev spaces $(0<q<1)$. The relation between the smoothness imposed on functions and the rate of almost everywhere convergence of the generalized Bochner-Riesz means is given.
In this paper, we introduce the statistically multiplicative convergent sequences in locally solid Riesz algebras with respect to the algebra multiplication and the solid topology. We study on this concept and we give the notion of $\mathbb{st_m}$-bounded sequence, and also, we prove some relations between this convergence and the other convergences such as the order convergence and the statistical convergence in topological spaces. Also, we give some results related to semiprime $f$-algebras.
Let $(E,τ)$ be a locally solid vector lattice. A filter $\mathcal{F}$ on the set $E$ is said to be converge to a vector $e\in E$ if, each zero neighborhood set $U$ containing $e$, $U$ belongs to $\mathcal{F}$. We study on the concept of this convergence and give some basic properties of it.
A sequence $(x_n)$ in a lattice-normed space $(X,p,E)$ is statistical $p$-convergent to $x\in X$ if there exists a statistical $p$-decreasing sequence $q\stpd 0$ with an index set $K$ such that $δ(K)=1$ and $p(x_{n_k}-x)\leq q_{n_k}$ for every $n_k\in K$. This convergence has been investigated recently for $(X,p,E)=(E,|\cdot|,E)$ under the name of statistical order convergence and under the name of statistical multiplicative order convergence, and also, for taking $E$ as a locally solid Riesz space under the names statistically unbounded $τ$-convergence and statistically multiplicative converg…
In this paper, we develop a thorough analysis of the boundedness properties of the maximal operator for the Bochner-Riesz means related to the Fourier-Bessel expansions. For this operator, we study weighted and unweighted inequalities in the spaces L^p((0,1),x^{2ν+1}dx). Moreover, weak and restricted weak type inequalities are obtained for the critical values of p. As a consequence, we deduce the almost everywhere pointwise convergence of these means.
In this paper, we investigate operators on Riesz algebras, which are continuous with respect to multiplicative modifications of order convergence and relatively uniform convergence. We also introduce and study mo-Lebesgue, mo-$KB$, and mo-Levi operators.
A sequence $(x_n)$ in a locally solid Riesz space $(E,τ)$ is said to be statistically unbounded $τ$-convergent to $x\in E$ if, for every zero neighborhood $U$, $\frac{1}{n}\big\lvert\{k\leq n:\lvert x_k-x\rvert\wedge u\notin U\}\big\rvert\to 0$ as $n\to\infty$. In this paper, we introduce this concept and give the notions $st$-$u_τ$-closed subset, $st$-$u_τ$-Cauchy sequence, $st$-$u_τ$-continuous and $st$-$u_τ$-complete locally solid vector lattice. Also, we give some relations between the order convergence and the $st$-$u_τ$-convergence.
A linear operator $T$ between two lattice-normed locally solid Riesz spaces is said to be $p_τ$-continuous if, for any $p_τ$-null net $(x_α)$, the net $(Tx_α)$ is $p_τ$-null, and $T$ is also said to be $p_τ$-bounded operator if it sends $p_τ$-bounded subsets to $p_τ$-bounded subsets. They are generalize several known classes of operators such as continuous, order continuous, $p$-continuous, order bounded, $p$-bounded operators, etc. We also study $up_τ$-continuous operators between lattice-normed locally solids Riesz spaces.
Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, $T$, in [{{\sc Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson}, {Convergence in Riesz spaces with conditional expectation operators}, {\em Positivity}, {\bf 19} {(2015), 647-657}}] as $T$-strong convergence and convergence in $T$-conditional probability, respectively. Generalized $L^{p}$ spaces for the cases of $p=1,2,\infty$, were discussed in the setting of Riesz spaces as $\mathcal{L}^{p}(T)$ spaces in [{{\sc C. C. A. Labuschagne, B. A. Watson}, {Discrete…
We provide the conditions for the boundedness of the Bochner-Riesz operator acting between two different Grand Lebesgue Spaces. Moreover we obtain a lower estimate for the constant appearing in the Lebesgue-Riesz norm estimation of the Bochner-Riesz operator and we investigate the convergence of the Bochner-Riesz approximation in Lebesgue-Riesz spaces.
Let $X$ be a vector lattice and $(E,τ)$ be a locally solid vector lattice. An operator $T:X\to E$ is said to be $ob$-bounded if, for each order bounded set $B$ in $X$, $T(B)$ is topologically bounded in $E$. In this paper, we study on algebraic properties of $ob$-bounded operators with respect to the topology of uniform convergence and equicontinuous convergence.
Several recent papers investigated unbounded and statistical versions of order convergence and topology convergence in locally solid Riesz space. In this papers, we study the statistical unbounded order and topology convergence in Riesz spaces and Banach lattices. In particular, we study the relationship of those convergence and characterize order continuous, KB, reflexive Banach lattices in terms of these convergence.
We extend the Koopman-von Neumann convergence condition on the Cesàro mean to the context of a Dedekind complete Riesz space with weak order unit. As a consequence, a characterisation of conditional weak mixing is given in the Riesz space setting. The results are applied to convergence in $L^1$.
We prove an almost everywhere convergence result for Bochner-Riesz means of $L^p$ functions on Heisenberg-type groups, yielding the existence of a $p>2$ for which convergence holds for means of arbitrarily small order. The proof hinges on a reduction of weighted $L^2$ estimates for the maximal Bochner-Riesz operator to corresponding estimates for the non-maximal operator, and a `dual Sobolev trace lemma', whose proof is based on refined estimates for Jacobi polynomials.
In this paper, we study the $L^p$-Bochner-Riesz mean summability problem related to the spectrum of some particular Sturm-Liouville operators in the weighted $L^p([a,b],ω).$ Our purpose is to establish suitable conditions under which the Bochner-Riesz expansion of a function $f \in L^p([a,b],ω)$,$1<p<\infty$, in two generalisations of Slepian's basis, converges to $f$ in $L^p([a,b],ω)$.
Outer product frames are important objects in Hilbert space frame theory. But very little is known about them. In this paper, we make the first detailed study of the family of outer product frames induced directly by vector sequences. We are interested in both the quantitative attributs of these outer product sequences (in particular, their Riesz and frame bounds), as well as their independence and spanning properties. We show that Riesz sequences of vectors yield Riesz sequences of outer products with the same (or better) Riesz bounds. Equiangular tight frames are shown to produce Riesz seque…
The main purpose of this paper is to give a vector lattice version of a Theorem by Burkholder about convergence of martingales. The proof is based on a vector lattice analogue of Austin's sample function theorem, proved recently by Grobler, Labuschagne and Marraffa and on a new characterization of elements of the sup-completion of a universally complete vector lattice which do not belong to the space.
Let $E$ be a Riesz space and let $E^{\sim}$ denote its order dual. The orthomorphisms $Orth(E)$ on $E,$ and the ideal center $Z(E)$ of $E,$ are naturally embedded in $Orth(E^{\sim})$ and $Z(E^{\sim})$ respectively. We construct two unital algebra and order continuous Riesz homomorphisms \[ γ:((Orth(E))^{\sim})_{n}^{\sim}\rightarrow Orth(E^{\sim})\text{ }% \] and \[ m:Z(E)^{\prime\prime}\rightarrow Z(E^{\sim}) \] that extend the above mentioned natural inclusions respectively. Then, the range of $γ$ is an order ideal in $Orth(E^{\sim})$ if and only if $m$ is surjective. Furthermore, $m$ is surj…
We prove an abstract Fubini-type theorem in the context of monoidal and enriched category theory, and as a corollary we establish a Fubini theorem for integrals on arbitrary convergence spaces that generalizes (and entails) the classical Fubini theorem for Radon measures on compact Hausdorff spaces. Given a symmetric monoidal closed adjunction satisfying certain hypotheses, we show that an associated monad of natural distributions D is commutative. Applying this result to the monoidal adjunction between convergence spaces and convergence vector spaces, the commutativity of D amounts to a Fubin…
Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, T, in [Y. Azouzi, W.-C. Kuo, K. Ramdane, B. A. Watson, Convergence in Riesz spaces with conditional expectation operators, Positivity, 19 (2015), 647-657] as T-strong convergence and convergence in T- conditional probability, respectively. Generalized Lp spaces for the cases of p = 1; 2;1, were discussed in the setting of Riesz spaces as Lp(T) spaces in [C. C. A. Labuschagne, B. A. Watson, Discrete stochastic integration in Riesz spaces, Positivity, 14 (2010…
In this paper we study the L^p-convergence of the Riesz means for the sublaplacian on the sphere S^{2n-1} in the complex n-dimensional space C^n. We show that the Riesz means of order delta of a function f converge to f in L^p(S^{2n-1}) when delta>delta(p):=(2n-1)|1\2-1\p|. The index delta(p) improves the one found by Alexopoulos and Lohoue', $2n|1\2-1\p|$, and it coincides with the one found by Mauceri and, with different methods, by Mueller in the case of sublaplacian on the Heisenberg group.
Notions of convergence and continuity specifically adapted to Riesz ideals I of the space of continuous real-valued functions on a Lindelöf locally compact Hausdorff space are given, and used to prove Stone-Weierstraß-type theorems for I. As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on I are uniquely determined by their restriction to various point-separating subsets of I. A very special case of this is the characterization of the strong determinacy of moment problems, which is rederived here in a rather general setting and with…
Let $H = -Δ+ |x|^2$ be the Hermite operator in ${\mathbb R}^n$. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with $H$ which is defined by $S_R^λ(H)f(x) = \sum\limits_{k=0}^{\infty} \big(1-{2k+n\over R^2}\big)_+^λ P_k f(x).$ Here $P_k f$ is the $k$-th Hermite spectral projection operator. For $2\le p<\infty$, we prove that $$ \lim\limits_{R\to \infty} S_R^λ(H) f=f \ \ \ \text{a.e.} $$ for all $f\in L^p(\mathbb R^n)$ provided that $λ> λ(p)/2$ and $λ(p)=\max\big\{ n\big({1/2}-{1/p}\big)-{1/ 2}, \, 0\big\}.$ Conversely, we also show the convergence gen…
This article provides a deeper study of the Riesz transform commutators associated with the Neumann Laplacian operator $Δ_N$ on $\mathbb R^n$. Along the line of singular value estimates for Riesz transform commutators established by Janson--Wolff and Rochberg--Semmes, we establish a full range of Schatten-$p$ class characterization for these commutators.
Some sharp results related to the convergence of means and families of operators generated by the generalized Bochner-Riesz kernels are obtained. The exact order of approximation of functions by these methods via $K$-functional (or its realization in the case of the space $L_p$, $0<p<1$) is derived.
Given a frequency $λ=(λ_n)$, we study when almost all vertical limits of a $\mathcal{H}_1$-Dirichlet series $\sum a_n e^{-λ_ns}$ are Riesz-summable almost everywhere on the imaginary axis. Equivalently, this means to investigate almost everywhere convergence of Fourier series of $H_1$-functions on so-called $λ$-Dirichlet groups, and as our main technical tool we need to invent a weak-type $(1, \infty)$ Hardy-Littlewood maximal operator for such groups. Applications are given to $H_1$-functions on the infinite dimensional torus $\mathbb{T}^\infty$, ordinary Dirichlet series $\sum a_n n^{-s}$, a…
We study a capacity theory based on a definition of a Riesz potential in metric spaces with a doubling measure. In this general setting, we study the basic properties of the Riesz capacity, including monotonicity, countable subadditivity and several convergence results. We define a modified version of the Hausdorff measure and provide lower bound and upper bound estimates for the capacity in terms of the modified Hausdorff content.
We define bidual bounded $uo$-convergence in vector lattices and investigate relations between this convergence and $b$-property. We prove that for a regular Riesz dual system $\langle X,X^{\sim}\rangle$, $X$ has $b$-property if and only if the order convergence in $X$ agrees with the order convergence in $X^{\sim\sim}$.
We study the boundary behavior of functions in the Hardy spaces on the infinite dimensional polydisk. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz-Zygmund type theorems for radial convergence. As a consequence one obtains easy new proofs of the brothers F. and M. Riesz theorems in infinite dimension. Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.
In this paper we generalize the Hájek-Rényi-Chow maximal inequality for submartingales to $L^p$ type Riesz spaces with conditional expectation operators. As applications we obtain a submartingale convergence theorem and a strong law of large numbers in Riesz spaces. Along the way we develop a Riesz space variant of the Clarkson's inequality for $1\le p\le 2$.
It is not usual to characterize an operator valued series via completeness of multiplier spaces. In this study, by using a series of bounded linear operators, we introduce the space $M^\infty_{R}\big(\sum_k T_k\big)$ of Riesz summability which is a generalization of the Cesàro summability. Therefore, we give the completeness criteria of these spaces with $c_0(X)$-multiplier convergent operator series. It is a natural consequence that one can characterize the completeness of a normed space through $M^\infty_{R}\big(\sum_k T_k\big)$ which will be assumed that is complete for every $c_0(X)$-multi…
Various topics in stochastic processes have been considered in the abstract setting of Riesz spaces, for example martingales, martingale convergence, ergodic theory, AMARTS, Markov processes and mixingales. Here we continue the relaxation of conditional independence begun in the study of mixingales and study mixing processes. The two mixing coefficients which will be considered are the $α$ (strong) and $φ$ (uniform) mixing coefficients. We conclude with mixing inequalities for these types of processes. In order to facilitate this development, the study of generalized $L^1$ and $L^\infty$ space…
In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace operator. The sufficient conditions for summability is obtained. For the orders of Riesz means, which greater than critical index $\frac{N-1}{2}$ we established the estimation for maximal operator of the Riesz means. Note that when order $α$ of Riesz means is less than critical index then for establish of the almost everywhere convergence requests to use other methods form proving negative results. We have constructed different method of summability of Laplace series,…
A net $(x_γ)_{γ\inΓ}$ in a locally solid Riesz space $(X,τ)$ is said to be unbounded $τ$-convergent to $x$ if $|x_γ-x|\wedge u\mathop{\oversetτ{\longrightarrow}} 0$ for all $u\in X_+$. We recall that there is a locally solid linear topology $\mathfrak{u}τ$ on $X$ such that unbounded $τ$-convergence coincides with $\mathfrak{u}τ$-convergence, and moreover, $\mathfrak{u}τ$ is characterised as the weakest locally solid linear topology which coincides with $τ$ on all order bounded subsets. It is with this motivation that we introduce, for a uniform lattice $(L,u)$, the weakest lattice uniformity $…
A Gabor system generated by a window function $g\in L^2(\mathbb{R}^d)$ and a separable set $Λ\times Γ\subset \mathbb{R}^{2d}$ is the collection of time-frequency shifts of $g$ given by $\mathcal G(g, Λ\times Γ) = \left\{ e^{2πi ξ\cdot t}g(t-x)\right\}_{ (x,ξ)\in Λ\times Γ}$. One of the fundamental problems in Gabor analysis is to characterize all windows and time-frequency sets that generate a Gabor frame or Gabor orthonormal basis. The case of Gabor orthonormal bases generated by characteristic functions $g=χ_Ω$ has been solved by Han and Wang. In this paper, we build on these results and obt…
For appropriate orthonormal wavelet basis $\{ψ_{j\,k}^e \}_{j\in\mathbb{Z}\,k\in\mathbb{Z}^d\,e\in\{0,1\}^d}$, constants $p$ and $γ$, if $\mathcal{I}_γ$ denotes the Riesz fractional integral operator of order $γ$ and $(η_{j\,k\,e})_{j\in\mathbb{Z} k\in\mathbb{Z}^d \,e\in\{0,1\}^d}$ a sequence of independent identically distributed symmetric $p$-stable random variables, we investigate the convergence of the series $\sum\limits_{j\,k\,e} η_{j\,k\,e} \mathcal{I}_γ ψ_{j\,k\,}^e$. Similar results are also studied for modified fractional integral operators. Finally, some geometric properties related…
Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$. In this article, the authors establish the Riesz transform characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$, for $w\in A_{q}(\mathbb{R}^n)$ and $w^{-1}\in A_{2-\frac{2}{n}}(\mathbb{R}^n)$ with $n\geq 3$, $q\in[1,2]$ and $p\in(q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]$ if, for some $r\in[1,\,2)$, $\{tL_w e^{-tL_w}\}_{t\geq 0}$ satisfies the weighted $L^r-L^2$ full off-diagonal estimate…
If $\{x_n\}_{n \in \mathbb{N}}$ is a frame for a Hilbert space $H,$ then there exists a canonical dual frame $\{\tilde{x_n}\}_{n \in \mathbb{N}}$ such that for every $x \in H$ we have $x = \sum \langle x, \tilde{x_n} \rangle \, x_n,$ with unconditional convergence of this series. However, if the frame is not a Riesz basis, then there exist alternative duals $\{y_n\}_{n \in \mathbb{N}}$ and synthesis-pseudo duals $\{z_n\}_{n \in \mathbb{N}}$ such that $x = \sum \langle x, y_n \rangle \, x_n,$ and $x = \sum \langle x, x_n \rangle \, z_n,$ for every $x.$ We characterize the frames for which the f…
