On the existence of distinguished bases in a $V$-space

Kung-Wei Yang

Displaying similar documents to “On the existence of distinguished bases in a $V$ -space”

On bases and unconditional bases in the spaces $L^{p} (d μ)$ ,1≤p<∞

K. Kazarian (1982)

Studia Mathematica

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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 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...

Positive bases in ordered subspaces with the Riesz decomposition property

Vasilios Katsikis, Ioannis A. Polyrakis (2006)

Studia Mathematica

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In this article we suppose that E is an ordered Banach space whose positive cone is defined by a countable family $ℱ = f_{i} | i \in ℕ$ of positive continuous linear functionals on E, i.e. E₊ = x ∈ E | $f_{i} (x) \geq 0$ for each i, and we study the existence of positive (Schauder) bases in ordered subspaces X of E with the Riesz decomposition property. We consider the elements x of E as sequences $x = (f_{i} (x))$ and we develop a process of successive decompositions of a quasi-interior point of X₊ which at each step gives elements with...

A construction of basis in $C^{(1)} (I^{2})$

Z. Ciesielski (1969)

Studia Mathematica

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Bases in the spaces $C$ and $L^{1}$

S. J. Szarek (1979-1980)

Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")

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A simple proof of two theorems concerning bases of $C (0, 1)$

G. Schechtman (1978-1979)

Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")

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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 < +∞).

Uniqueness of unconditional bases in $c_{0}$ -products

P. Casazza, N. Kalton (1999)

Studia Mathematica

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We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does $c_{0} (X)$ . We also give some positive results including a simpler proof that $c_{0} (ℓ_{1})$ has a unique unconditional basis and a proof that $c_{0} (ℓ_{p_{n}}^{N_{n}})$ has a unique unconditional basis when $p_{n} ￬ 1$ , $N_{n + 1} \geq 2 N_{n}$ and $(p_{n} - p_{n + 1}) l o g N_{n}$ remains bounded.

Greedy approximation and the multivariate Haar system

A. Kamont, V. N. Temlyakov (2004)

Studia Mathematica

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We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis in $L_{p} ([0, 1])$ , 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis...

Quasi-greedy bases and Lebesgue-type inequalities

S. J. Dilworth, M. Soto-Bajo, V. N. Temlyakov (2012)

Studia Mathematica

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We study Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. We mostly concentrate on the $L_{p}$ spaces. The novelty of the paper is in obtaining better Lebesgue-type inequalities under extra assumptions on a quasi-greedy basis than known Lebesgue-type inequalities for quasi-greedy bases. We consider uniformly bounded quasi-greedy bases of $L_{p}$ , 1 < p < ∞, and prove that for such bases an extra multiplier in the Lebesgue-type inequality can be taken...

On the existence of almost greedy bases in Banach spaces

S. J. Dilworth, N. J. Kalton, Denka Kutzarova (2003)

Studia Mathematica

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We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space X has a basis and contains a complemented subspace with a symmetric basis and finite cotype then...

Unconditionality of general Franklin systems in $L^{p} [0, 1]$ , 1 < p < ∞

Gegham G. Gevorkyan, Anna Kamont (2004)

Studia Mathematica

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By a general Franklin system corresponding to a dense sequence = (tₙ, n ≥ 0) of points in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots , that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is that each general Franklin system is an unconditional basis in $L^{p} [0, 1]$ , 1 < p < ∞.

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.

On a result of Olevskiĭ : a uniformly bounded orthonormal sequence is not a basis for $C [0, 1]$

A. Pełczynski (1973-1974)

Séminaire Analyse fonctionnelle (dit "Maurey-Schwartz")

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On unconditional polynomial bases of the space $L_{p}$

Z. Chanturia (1981)

Studia Mathematica

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On the non-equivalence of rearranged Walsh and trigonometric systems in $L_{p}$

Aicke Hinrichs, Jörg Wenzel (2003)

Studia Mathematica

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We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in $L_{p}$ for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.

Second derivatives of norms and contractive complementation in vector-valued spaces

Bas Lemmens, Beata Randrianantoanina, Onno van Gaans (2007)

Studia Mathematica

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We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $ℓ_{p} (X)$ , where X is a Banach space with a 1-unconditional basis and p ∈ (1,2) ∪ (2,∞). If the norm of X is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of $ℓ_{p} (X)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection...

Haar wavelets on the Lebesgue spaces of local fields of positive characteristic

Biswaranjan Behera (2014)

Colloquium Mathematicae

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We construct the Haar wavelets on a local field K of positive characteristic and show that the Haar wavelet system forms an unconditional basis for $L^{p} (K)$ , 1 < p < ∞. We also prove that this system, normalized in $L^{p} (K)$ , is a democratic basis of $L^{p} (K)$ . This also proves that the Haar system is a greedy basis of $L^{p} (K)$ for 1 < p < ∞.

Displaying similar documents to “On the existence of distinguished bases in a V -space”

Displaying similar documents to “On the existence of distinguished bases in a $V$ -space”