Positive bases in ordered subspaces with the Riesz decomposition property

Vasilios Katsikis; Ioannis A. Polyrakis

Displaying similar documents to “Positive bases in ordered subspaces with the Riesz decomposition property”

On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems

Robert E. Zink (2002)

Colloquium Mathematicae

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In one of the earliest monographs that involve the notion of a Schauder basis, Franklin showed that the Gram-Schmidt orthonormalization of a certain Schauder basis for the Banach space of functions continuous on [0,1] is again a Schauder basis for that space. Subsequently, Ciesielski observed that the Gram-Schmidt orthonormalization of any Schauder system is a Schauder basis not only for C[0,1], but also for each of the spaces $L^{p} [0, 1]$ , 1 ≤ p < ∞. Although perhaps not probable, the latter...

On the Fejér-F. Riesz inequality in $L^{p}$

Yoram Sagher (1977)

Studia Mathematica

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Generalized Riesz products produced from orthonormal transforms

Nikolaos Atreas, Antonis Bisbas (2012)

Colloquium Mathematicae

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Let $ℳ_{p} = {m_{k}}_{k = 0}^{p - 1}$ be a finite set of step functions or real valued trigonometric polynomials on = [0,1) satisfying a certain orthonormality condition. We study multiscale generalized Riesz product measures μ defined as weak-* limits of elements $μ_{N} \in V_{N} (N \in ℕ)$ , where $V_{N}$ are $p^{N}$ -dimensional subspaces of L₂() spanned by an orthonormal set which is produced from dilations and multiplications of elements of $ℳ_{p}$ and $\bar{⋃_{N \in ℕ} V_{N}} = L ₂ ()$ . The results involve mutual absolute continuity or singularity of such Riesz products extending previous...

Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces

Aydin Sh. Shukurov (2012)

Colloquium Mathematicae

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A necessary condition for Kostyuchenko type systems and system of powers to be a basis in $L_{p}$ (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in $L_{p}$ (1 ≤ p < +∞).

Dichotomy of global density of Riesz capacity

Hiroaki Aikawa (2016)

Studia Mathematica

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Let $C_{α}$ be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density $̲ (C_{α}, E, r) = i n f_{x \in ℝ ⁿ} C_{α} (E \cap B (x, r)) / C_{α} (B (x, r))$ for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that $l i m_{r \to \infty} ̲ (C_{α}, E, r)$ is either 0 or 1; the first case occurs if and only if $̲ (C_{α}, E, r)$ is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also...

Addendum to "Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces" (Colloq. Math. 127 (2012), 105-109)

Aydin Sh. Shukurov (2014)

Colloquium Mathematicae

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It is well known that if φ(t) ≡ t, then the system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is a basis in some Lebesgue space $L_{p}$ . The aim of this short note is to show that the answer to this question is negative.

The universal Banach space with a $K$ -suppression unconditional basis

Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2018)

Commentationes Mathematicae Universitatis Carolinae

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Using the technique of Fraïssé theory, for every constant $K \geq 1$ , we construct a universal object $𝕌_{K}$ in the class of Banach spaces possessing a normalized $K$ -suppression unconditional Schauder basis.

The basis property in $L_{p}$ of the boundary value problem rationally dependent on the eigenparameter

N. B. Kerimov, Y. N. Aliyev (2006)

Studia Mathematica

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We consider a Sturm-Liouville operator with boundary conditions rationally dependent on the eigenparameter. We study the basis property in $L_{p}$ of the system of eigenfunctions corresponding to this operator. We determine the explicit form of the biorthogonal system. Using this we establish a theorem on the minimality of the part of the system of eigenfunctions. For the basisness in L₂ we prove that the system of eigenfunctions is quadratically close to trigonometric systems. For the basisness...

Subsystems of the Schauder system whose orthonormalizations are Schauder bases for $L^{p} [0, 1]$

Robert E. Zink (1989)

Colloquium Mathematicae

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On some subspaces of Morrey-Sobolev spaces and boundedness of Riesz integrals

Mouhamadou Dosso, Ibrahim Fofana, Moumine Sanogo (2013)

Annales Polonici Mathematici

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For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^{q}, l^{p})}^{α}$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^{q}, l^{p})$ and contains the Lebesgue space $L^{α}$ . We study the closure ${(L^{q}, l^{p})}_{c, 0}^{α}$ in ${(L^{q}, l^{p})}^{α}$ of the space of test functions (infinitely differentiable and with compact support in $ℝ^{d}$ ) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W ¹ ({(L^{q}, l^{p})}^{α})$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...

Three-space problems and bounded approximation properties

Wolfgang Lusky (2003)

Studia Mathematica

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Let ${R ₙ}_{n = 1}^{\infty}$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an $ℒ_{p}$ -space, then both X and A have bases. We apply these results to show that the spaces $C_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset C ()$ and $L_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset L ₁ ()$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.

Sharp inequalities for Riesz transforms

Adam Osękowski (2014)

Studia Mathematica

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We establish the following sharp local estimate for the family ${R_{j}}_{j = 1}^{d}$ of Riesz transforms on $ℝ^{d}$ . For any Borel subset A of $ℝ^{d}$ and any function $f : ℝ^{d} \to ℝ$ , $\int_{A} | R_{j} f (x) | d x \leq C_{p} {| | f | |}_{L^{p} (ℝ^{d})} {| A |}^{1 / q}$ , 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_{p} = {[2^{q + 2} Γ (q + 1) / π^{q + 1} \sum_{k = 0}^{\infty} {(- 1)}^{k} / {(2 k + 1)}^{q + 1}]}^{1 / q}$ , 1 < p < 2, and $C_{p} = {[4 Γ (q + 1) / π^{q} \sum_{k = 0}^{\infty} 1 / {(2 k + 1)}^{q}]}^{1 / q}$ , 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

Uniqueness of unconditional basis of $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ , 0 < p < 1

F. Albiac, C. Leránoz (2002)

Studia Mathematica

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We prove that the quasi-Banach spaces $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ (0 < p < 1) have a unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss and Tzafriri have previously proved that the same is true for the respective Banach envelopes $ℓ ₁ (c ₀)$ and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

Riesz potentials derived by one-mode interacting Fock space approach

Nobuhiro Asai (2007)

Colloquium Mathematicae

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The main aim of this short paper is to study Riesz potentials on one-mode interacting Fock spaces equipped with deformed annihilation, creation, and neutral operators with constants $c_{0, 0}, c_{1, 1} \in ℝ$ and $c_{0, 1} > 0$ , $c_{1, 2} \geq 0$ as in equations (1.4)-(1.6). First, to emphasize the importance of these constants, we summarize our previous results on the Hilbert space of analytic L² functions with respect to a probability measure on ℂ. Then we consider the Riesz kernels of order 2α, $α = c_{0, 1} / c_{1, 2}$ , on ℂ if $0 < c_{0, 1} < c_{1, 2}$ , which can be derived from...

A short proof on lifting of projection properties in Riesz spaces

Marek Wójtowicz (1999)

Commentationes Mathematicae Universitatis Carolinae

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Let $L$ be an Archimedean Riesz space with a weak order unit $u$ . A sufficient condition under which Dedekind [ $σ$ -]completeness of the principal ideal $A_{u}$ can be lifted to $L$ is given (Lemma). This yields a concise proof of two theorems of Luxemburg and Zaanen concerning projection properties of $C (X)$ -spaces. Similar results are obtained for the Riesz spaces $B_{n} (T)$ , $n = 1, 2, \dots$ , of all functions of the $n$ th Baire class on a metric space $T$ .

Corrigendum to the paper “The universal Banach space with a $K$ -suppression unconditional basis”

Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2020)

Commentationes Mathematicae Universitatis Carolinae

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We observe that the notion of an almost ${𝔉ℑ}_{K}$ -universal based Banach space, introduced in our earlier paper [1]: Banakh T., Garbulińska-Wegrzyn J., The universal Banach space with a $K$ -suppression unconditional basis, Comment. Math. Univ. Carolin. 59 (2018), no. 2, 195–206, is vacuous for $K = 1$ . Taking into account this discovery, we reformulate Theorem 5.2 from [1] in order to guarantee that the main results of [1] remain valid.

Orthosymmetric bilinear map on Riesz spaces

Elmiloud Chil, Mohamed Mokaddem, Bourokba Hassen (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $E$ be a Riesz space, $F$ a Hausdorff topological vector space (t.v.s.). We prove, under a certain separation condition, that any orthosymmetric bilinear map $T : E \times E \to F$ is automatically symmetric. This generalizes in certain way an earlier result by F. Ben Amor [On orthosymmetric bilinear maps, Positivity 14 (2010), 123–134]. As an application, we show that under a certain separation condition, any orthogonally additive homogeneous polynomial $P : E \to F$ is linearly represented. This fits in the type of...

L¹ representation of Riesz spaces

Bahri Turan (2006)

Studia Mathematica

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Let E be a Riesz space. By defining the spaces $L ¹_{E}$ and $L_{E}^{\infty}$ of E, we prove that the center $Z (L ¹_{E})$ of $L ¹_{E}$ is $L_{E}^{\infty}$ and show that the injectivity of the Arens homomorphism m: Z(E)” → Z(E˜) is equivalent to the equality $L ¹_{E} = Z {(E)}^{'}$ . Finally, we also give some representation of an order continuous Banach lattice E with a weak unit and of the order dual E˜ of E in $L ¹_{E}$ which are different from the representations appearing in the literature.

Variations on Bochner-Riesz multipliers in the plane

Daniele Debertol (2006)

Studia Mathematica

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We consider the multiplier $m_{μ}$ defined for ξ ∈ ℝ by $m_{μ} (ξ) ≐ {((1 - ξ ₁ ² - ξ ₂ ²) / (1 - ξ ₁))}^{μ} 1_{D} (ξ)$ , D denoting the open unit disk in ℝ. Given p ∈ ]1,∞[, we show that the optimal range of μ’s for which $m_{μ}$ is a Fourier multiplier on $L^{p}$ is the same as for Bochner-Riesz means. The key ingredient is a lemma about some modifications of Bochner-Riesz means inside convex regions with smooth boundary and non-vanishing curvature, providing a more flexible version of a result by Iosevich et al. [Publ. Mat. 46 (2002)]. As an application, we show...