Let $X$ and $Y$ be Banach spaces, and $T:X^*\to Y$ be an operator. We prove that if $X$ is Asplund and $Y$ has the approximation property, then for each Radon probability $μ$ on $(B_{X^*},w^*)$ there is a sequence of $w^*$-to-norm continuous operators $T_n:X^*\to Y$ such that $\|T_n(x^*)-T(x^*)\| \to 0$ for $μ$-a.e. $x^*\in B_{X^*}$; if $Y$ has the $λ$-bounded approximation property for some $λ\geq 1$, then the sequence can be chosen in such a way that $\|T_n\|\leq λ\|T\|$ for all $n\in \mathbb{N}$. The same conclusions hold if $X$ contains no subspace isomorphic to $\ell_1$, $Y$ has the approximation property (resp., $λ$-bounded approximation property) and $T$ has separable range. This extends to the non-separable setting a result by Mercourakis and Stamati.
Let $\I$ be an ideal on $\N$ which is either analytic or coanalytic. Assume that $(f_n)$ is a sequence of functions with the Baire property from a Polish space $X$ into a complete metric space $Z$, which is divergent on a comeager set. We investigate the Baire category of $\I$-convergent subsequences and rearrangements of $(f_n)$. Our result generalizes a theorem of Kallman. A similar theorem for subsequences is obtained if $(X,μ)$ is a $σ$-finite complete measure space and a sequence $(f_n)$ of measurable functions from $X$ to $Z$ is $\I$-divergent $μ$-almost everywhere. Then the set of subse…
The spaces $W_α$ are the Banach spaces whose duals are isometric to $\ell_1$ and such that the standard basis of $\ell_1$ is $w^*$-convergent to $α\in \ell_1$. The core result of our paper proves that an $\ell_1$-predual $X$ contains isometric copies of all $W_α$, where the norm of $α$ is controlled by the supremum of the norms of the $w^*$-cluster points of the extreme points of the closed unit ball in $\ell_1$. More precisely, for every $\ell_1$-predual $X$ we have $$ r^*(X)=\sup\left\lbrace \left\|g^*\right\|: g^*\in \left(\mathrm{ext}\, B_{\ell_1}\right)'\right\rbrace =\sup \left\lbrace \l…
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.
A net $(x_α)$ in a vector lattice $X$ is said to be {unbounded order convergent} (or uo-convergent, for short) to $x\in X$ if the net $(\abs{x_α-x}\wedge y)$ converges to 0 in order for all $y\in X_+$. In this paper, we study unbounded order convergence in dual spaces of Banach lattices. Let $X$ be a Banach lattice. We prove that every norm bounded uo-convergent net in $X^*$ is $w^*$-convergent iff $X$ has order continuous norm, and that every $w^*$-convergent net in $X^*$ is uo-convergent iff $X$ is atomic with order continuous norm. We also characterize among $σ$-order complete Banach lattic…
The main result of this paper is a proof that, for any $f \in L_1[a,b]$, a sequence of its orthogonal projections $(P_{Δ_n}(f))$ onto splines of order $k$ with arbitrary knots $Δ_n$, converges almost everywhere provided that the mesh diameter $|Δ_n|$ tends to zero, namely \[ f \in L_1[a,b] \Rightarrow P_{Δ_n}(f,x) \to f(x) \quad \mbox{a.e.} \quad (|Δ_n|\to 0)\,. \] This extends the earlier result that, for $f \in L_p$, we have convergence $P_{Δ_n}(f) \to f$ in the $L_p$-norm for $1 \le p \le \infty$.}
Fuzzy logic was introduced by Zadeh in 1965. Since then, the importance of fuzzy logic has come increasingly to the present.There are many applications of fuzzy logic in the field of science and engineering, e.g. population dynamics (Barros), chaos control (Feng,Fradkov), computer programming (Giles), nonlinear dynamical systems (Hong), etc. The concept of intuitionistic fuzzy set, as a generalization of fuzzy logic, was introduced by Atanassov in 1986. Quite recently Park has introduced the concept of intuitionistic fuzzy metric space, and Saadati and Park studied the notion of intuitionistic…
The main purpose of this paper is to introduce lacunary strong geometric zweier convergent sequence spaces $N_{θ}^{0} \left[Z\left(G\right)\right]$, $N_{θ} \left[Z\left(G\right)\right]$, $N_{θ}^{\infty } \left[Z\left(G\right)\right]$consisting of all sequences $x=\left(x_{k} \right)$such that $\left[Z\left(G\right)\right]x$ are in the spaces $N_{θ}^{0} ,N_{θ} {\rm and\; }N_{θ}^{\infty } $ respectively, which are normed. We prove certain topological properties of these spaces and compute their lacunary stastical zweier convergence.
In this paper, we study stability of $M$-compactness for $l^p$ sum of Banach spaces for $1\leq p<\infty$. We also obtain a characterization of $M$-compact sets in terms of statistically maximizing sequence, a notion which is weaker than a maximizing sequence. Moreover, we introduce the notion of $\mathcal{I}$-$M$-compactness of a bounded subset $M$ of a normed linear space $X$ with respect to an ideal $\mathcal{I}$ and show that it is equivalent to $M$-compactness for non-trivial admissible ideals.
Necessary and sufficient oscillation conditions are given for a weakly convergent sequence (resp. relatively weakly compact set) in the Bochner-Lebesgue space $\l1$ to be norm convergent (resp. relatively norm compact), thus extending the known results for $\rl1$. Similarly, necessary and sufficient oscillation conditions are given to pass from weak to limited (and also to Pettis-norm) convergence in $\l1$. It is shown that tightness is a necessary and sufficient condition to pass from limited to strong convergence. Other implications between several modes of convergence in $\l1$ are also stud…
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…
We address an open problem posed recently by Almeida and Hästö in \cite{AlHa10}. They defined the spaces $\ellqp$ of variable integrability and summability and showed that $\|\cdot|\ellqp\|$ is a norm if $q$ is constant almost everywhere or if $\esssup_{x\in\R^n}1/p(x)+1/q(x)\le 1$. Nevertheless, the natural conjecture (expressed also in \cite{AlHa10}) is that the expression is a norm if $p(x),q(x)\ge 1$ almost everywhere. We show, that $\|\cdot|\ellqp\|$ is a norm, if $1\le q(x)\le p(x)$ for almost every $x\in\R^n.$ Furthermore, we construct an example of $p(x)$ and $q(x)$ with $\min(p(x),q(x…
Given a sequence $(X_n)$ of real or complex random variables and a sequence of numbers $(a_n)$, an interesting problem is to determine the conditions under which the series $\sum_{n=1}^\infty a_n X_n$ is almost surely convergent. This paper extends the classical Menshov--Rademacher theorem on the convergence of orthogonal series to general series of dependent random variables and derives interesting sufficient conditions for the almost everywhere convergence of trigonometric series with respect to singular measures whose Fourier transform decays to 0 at infinity with positive rate.
A Banach space $\X$ has the complete continuity property (CCP) if each bounded linear operator from $L_1$ into $\X$ is completely continuous (i.e., maps weakly convergent sequences to norm convergent sequences). The main theorem shows that a Banach space failing the CCP (resp., failing the CCP and failing cotype) has a subspace with a finite dimensional decomposition (resp., basis) which fails the CCP.
The recently developed notion of action convergence by Backhausz and Szegedy unifies and generalises the dense (graphon) and local-global (graphing) convergences of graph sequences. This is done through viewing graphs as operators and examining their dynamical properties. Suppose $(A_n)_n^\infty$ is a sequence of operators representing graphs, Cauchy with respect to the action metric. If $(A_n)_n^\infty$ has uniformly bounded $(p,q)$-norms where $(p,q)$ is any pair in $[1,\infty)\times(1,\infty)$, then Backhausz and Szegedy prove that $(A_n)_n^\infty$ has a limit operator which, moreover, must…
A Banach space $X$ is said to have property ($μ^s$) if every weak$^*$-null sequence in $X^*$ admits a subsequence such that all of its subsequences are Cesàro convergent to $0$ with respect to the Mackey topology. This is stronger than the so-called property (K) of Kwapień. We prove that property $(μ^s)$ holds for every subspace of a Banach space which is strongly generated by an operator with Banach-Saks adjoint (e.g. a strongly super weakly compactly generated space). The stability of property $(μ^s)$ under $\ell^p$-sums is discussed. For a family $\mathcal{A}$ of relatively weakly compact s…
In this paper, we show that the expansions of functions from $L^p$-Paley-Wiener type spaces in terms of the prolate spheroidal wave functions converge almost everywhere for $1<p<\infty$, even in the cases when they might not converge in $L^p$-norm. We thereby consider the classical Paley-Wiener spaces $PW_c^p\subset L^p(\mathcal{R})$ of functions whose Fourier transform is supported in $[-c,c]$ and Paley-Wiener like spaces $B_{α,c}^p\subset L^p(0,\infty)$ of functions whose Hankel transform $\mathcal{H}^α$ is supported in $[0,c]$.As a side product, we show the continuity of the projection oper…
In this paper we collect results concerning the {operator-norm} convergent {Trotter} product formula for two semigroups $\{\e^{- t A}\}_{t\geq 0}$, $\{\e^{- t B}\}_{t\geq 0}$, with densely defined generators $A$ and $B$ in a {Banach} space. Although the {strong} convergence in Banach space for contraction semigroups is known since the seminal paper by Trotter (1959), which after more than three decades was extended to convergence in the {operator-norm} topology in {Hilbert} spaces by Rogava (1993), the {operator-norm} convergence in a {Banach} space was established only in (2001).
Most recently, some new double sequence spaces $B(\mathcal{M}_{u})$, $B(\mathcal{C}_{\vartheta})$ where $\vartheta=\{b,bp,r,f,f_0\}$ and $B(\mathcal{L}_{q})$ for $0<q<\infty$ have been introduced as four-dimensional generalized difference matrix $B(r,s,t,u)$ domain on the double sequence spaces $\mathcal{M}_{u}$, $\mathcal{C}_{\vartheta}$ where $\vartheta=\{b,bp,r,f,f_0\}$ and $\mathcal{L}_{q}$ for $0<q<\infty$, and some topological properties, dual spaces, some new four-dimensional matrix classes and matrix transformations related to these spaces have also been studied by Tuğ and Başar and Tu…
In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e.~pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in $L^1$-norm, are analyzed. These findings extend and complement a number of earlier results by several authors.
Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type, where the Lebesgue measure is replaced by an arbitrary measure $μ$. In particular we show that the differentiability properties of Lipschitz functions at $μ$-almost every point are related to the decompositions of $μ$ in terms of rectifiable one-dimensional measures. As a consequence we obtain a differentiability result for Lipschitz functions with respect …
It is well-known that every weakly convergent sequence in $\ell_1$ is convergent in the norm topology (Schur's lemma). Phillips' lemma asserts even more strongly that if a sequence $(μ_n)_{n\in\mathbb N}$ in $\ell_\infty'$ converges pointwise on $\{0,1\}^\mathbb N$ to $0$, then its $\ell_1$-projection converges in norm to $0$. In this note we show how the second category version of Schur's lemma, for which a short proof is included, can be used to replace in Phillips' lemma $\{0,1\}^\mathbb N$ by any of its subsets which contains all finite sets and having some kind of interpolation property f…
