The haar system as an unconditional basis in Lp[0, 1]

1974 ◽  
Vol 15 (2) ◽  
pp. 108-111 ◽  
Author(s):  
V. F. Gaposhkin
Keyword(s):  
Unconditional Basis ◽  
Haar System

Some Sharp Estimates for the Haar System and Other Bases In $L^1(0,1)$

2014 ◽  
Vol 115 (1) ◽  
pp. 123
Author(s):  
Adam Osȩkowski
Keyword(s):  
Nonnegative Integer ◽  
Unconditional Basis ◽  
Haar System ◽  
Bellman Function ◽  
Real Numbers ◽  
Best Constant ◽  
Sharp Estimates

Let $h=(h_k)_{k\geq 0}$ denote the Haar system of functions on $[0,1]$. It is well known that $h$ forms an unconditional basis of $L^p(0,1)$ if and only if $1<p<\infty$, and the purpose of this paper is to study a substitute for this property in the case $p=1$. Precisely, for any $\lambda>0$ we identify the best constant $\beta=\beta_h(\lambda)\in [0,1]$ such that the following holds. If $n$ is an arbitrary nonnegative integer and $a_0$, $a_1$, $a_2$, $\ldots$, $a_n$ are real numbers such that $\bigl\|\sum_{k=0}^n a_kh_k\bigr\|_1\leq 1$, then \[ \Bigl|\Bigl\{x\in [0,1]:\Bigl|\sum_{k=0}^n \varepsilon_ka_kh_k(x)\Bigr|\geq \lambda\Bigr\}\Bigr|\leq \beta, \] for any sequence $\varepsilon_0, \varepsilon_1, \varepsilon_2,\ldots, \varepsilon_n$ of signs. A related bound for an arbitrary basis of $L^1(0,1)$ is also established. The proof rests on the construction of the Bellman function corresponding to the problem.

scholarly journals Hyperbolic Wavelet Analysis of Classical Isotropic and Anisotropic Besov–Sobolev Spaces

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Martin Schäfer ◽  
Tino Ullrich ◽  
Béatrice Vedel
Keyword(s):  
Haar System ◽  
A Priori ◽  
Natural Setting ◽  
The Novel ◽  
Major Question ◽  
Vanishing Moments ◽  
Classical Setting ◽  
Hyperbolic Wavelets ◽  
The Relationship ◽  
Mixed Smoothness

AbstractIn this paper we introduce new function spaces which we call anisotropic hyperbolic Besov and Triebel-Lizorkin spaces. Their definition is based on a hyperbolic Littlewood-Paley analysis involving an anisotropy vector only occurring in the smoothness weights. Such spaces provide a general and natural setting in order to understand what kind of anisotropic smoothness can be described using hyperbolic wavelets (in the literature also sometimes called tensor-product wavelets), a wavelet class which hitherto has been mainly used to characterize spaces of dominating mixed smoothness. A centerpiece of our present work are characterizations of these new spaces based on the hyperbolic wavelet transform. Hereby we treat both, the standard approach using wavelet systems equipped with sufficient smoothness, decay, and vanishing moments, but also the very simple and basic hyperbolic Haar system. The second major question we pursue is the relationship between the novel hyperbolic spaces and the classical anisotropic Besov–Lizorkin-Triebel scales. As our results show, in general, both approaches to resolve an anisotropy do not coincide. However, in the Sobolev range this is the case, providing a link to apply the newly obtained hyperbolic wavelet characterizations to the classical setting. In particular, this allows for detecting classical anisotropies via the coefficients of a universal hyperbolic wavelet basis, without the need of adaption of the basis or a-priori knowledge on the anisotropy.

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

Uniqueness of unconditional basis in quasi-Banach spaces which are not sufficiently Euclidean

Positivity ◽  
2010 ◽  
Vol 14 (4) ◽  
pp. 579-584 ◽  
Author(s):  
Fernando Albiac ◽  
Camino Leránoz
Keyword(s):  
Banach Spaces ◽  
Unconditional Basis

scholarly journals In what spaces is every closed normal cone regular?

1970 ◽  
Vol 17 (2) ◽  
pp. 121-125 ◽  
Author(s):  
C. W. McArthur
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Normal Cone ◽  
Closed Subspace ◽  
Locally Convex Space ◽  
Regular Part ◽  
Fréchet Space ◽  
Sequential Completeness ◽  
Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

scholarly journals TENSOR EXTENSION PROPERTIES OF C(K)-REPRESENTATIONS AND APPLICATIONS TO UNCONDITIONALITY

2010 ◽  
Vol 88 (2) ◽  
pp. 205-230 ◽  
Author(s):  
CHRISTOPH KRIEGLER ◽  
CHRISTIAN LE MERDY
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unconditional Basis ◽  
Compact Set ◽  
Bounded Operators ◽  
Space Setting ◽  
Uniformly Bounded

AbstractLet K be any compact set. The C*-algebra C(K) is nuclear and any bounded homomorphism from C(K) into B(H), the algebra of all bounded operators on some Hilbert space H, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept ofR-boundedness. Then we apply these results to operators with a uniformly bounded H∞-calculus, as well as to unconditionality on Lp. We show that any unconditional basis on Lp ‘is’ an unconditional basis on L2 after an appropriate change of density.

On the divergence of Fourier series in the general Haar system

2021 ◽  
pp. 1-10
Author(s):  
Martin Grigoryan ◽  
Artavazd Maranjyan
Keyword(s):  
Fourier Series ◽  
Measurable Function ◽  
Haar System ◽  
Bounded Measurable Function

For any countable set $D \subset [0,1]$, we construct a bounded measurable function $f$ such that the Fourier series of $f$ with respect to the regular general Haar system is divergent on $D$ and convergent on $[0,1]\backslash D$.

scholarly journals James quasi reflexive space has the fixed point property

1989 ◽  
Vol 39 (1) ◽  
pp. 25-30 ◽  
Author(s):  
M.A. Khamsi
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unconditional Basis ◽  
Fixed Point Property ◽  
Point Property ◽  
Reflexive Space ◽  
Lin's Method ◽  
Lin’S Method

We prove that the classical sequence James space has the fixed point property. This gives an example of Banach space with a non-unconditional basis where the Maurey-Lin's method applies.

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