Inequivalence of Wavelet Systems in $L₁(ℝ^d)$ and $BV(ℝ^d)$

Paweł Bechler

Displaying similar documents to “Inequivalence of Wavelet Systems in $L ₁ (ℝ^{d})$ and $B V (ℝ^{d})$ ”

Good-λ inequalities for wavelets of compact support

Sarah V. Cook (2004)

Colloquium Mathematicae

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For a wavelet ψ of compact support, we define a square function $S_{w}$ and a maximal function NΛ. We then obtain the $L_{p}$ equivalence of these functions for 0 < p < ∞. We show this equivalence by using good-λ inequalities.

Refinement type equations: sources and results

Rafał Kapica, Janusz Morawiec (2013)

Banach Center Publications

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It has been proved recently that the two-direction refinement equation of the form $f (x) = \sum_{n \in} c_{n, 1} f (k x - n) + \sum_{n \in ℤ} c_{n, - 1} f (- k x - n)$ can be used in wavelet theory for constructing two-direction wavelets, biorthogonal wavelets, wavelet packages, wavelet frames and others. The two-direction refinement equation generalizes the classical refinement equation $f (x) = \sum_{n \in ℤ} c ₙ f (k x - n)$ , which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation $f (x) = \int_{ℝ} c (y) f (k x - y) d y$ has also various interesting...

Local means and wavelets in function spaces

Hans Triebel (2008)

Banach Center Publications

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The paper deals with local means and wavelet bases in weighted and unweighted function spaces of type $B_{p q}^{s}$ and $F_{p q}^{s}$ on ℝⁿ and on ⁿ.

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 < ∞.

Asymptotic behaviour of Besov norms via wavelet type basic expansions

Anna Kamont (2016)

Annales Polonici Mathematici

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J. Bourgain, H. Brezis and P. Mironescu [in: J. L. Menaldi et al. (eds.), Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, 439-455] proved the following asymptotic formula: if $Ω \subset ℝ^{d}$ is a smooth bounded domain, 1 ≤ p < ∞ and $f \in W^{1, p} (Ω)$ , then $l i m_{s ↗ 1} (1 - s) \int_{Ω} \int_{Ω} {(| f (x) - f (y) |}^{p} {) / (| | x - y | |}^{d + s p}) d x d y = K \int_{Ω} {| \nabla f (x) |}^{p} d x$ , where K is a constant depending only on p and d. The double integral on the left-hand side of the above formula is an equivalent seminorm in the Besov space $B_{p}^{s, p} (Ω)$ . The purpose of this paper is to obtain analogous asymptotic formulae...

Decomposition systems for function spaces

G. Kyriazis (2003)

Studia Mathematica

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Let $Θ : = θ_{I}^{e} : e \in E, I \in D$ be a decomposition system for $L ₂ (ℝ^{d})$ indexed over D, the set of dyadic cubes in $ℝ^{d}$ , and a finite set E, and let $Θ ̃ : = Θ ̃_{I}^{e} : e \in E, I \in D$ be the corresponding dual functionals. That is, for every $f \in L ₂ (ℝ^{d})$ , $f = \sum_{e \in E} \sum_{I \in D} ⟨ f, Θ ̃_{I}^{e} ⟩ θ_{I}^{e}$ . We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨ f, Θ ̃_{I}^{e} ⟩$ , e ∈ E, I ∈ D. Typical examples of such decomposition...

A note on integer translates of a square integrable function on ℝ

Maciej Paluszyński (2010)

Colloquium Mathematicae

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We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${ψ (\cdot - n)}_{n = - \infty}^{\infty}$ , which is equivalent to the weight function $\sum_{n = - \infty}^{\infty} | ψ ̂ (\cdot + n) | ²$ being > 0 a.e.

Embeddings of Besov-Morrey spaces on bounded domains

Dorothee D. Haroske, Leszek Skrzypczak (2013)

Studia Mathematica

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We study embeddings of spaces of Besov-Morrey type, $i d_{Ω} :_{p ₁, u ₁, q ₁}^{s ₁} (Ω) ↪_{p ₂, u ₂, q ₂}^{s ₂} (Ω)$ , where $Ω \subset ℝ^{d}$ is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of $i d_{Ω}$ . This continues our earlier studies relating to the case of $ℝ^{d}$ . Moreover, we also characterise embeddings into the scale of $L_{p}$ spaces or into the space of bounded continuous functions.

Polar wavelets and associated Littlewood-Paley theory

Epperson Jay, Frazier Michael

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Abstract We develop an almost orthogonal wavelet-type expansion in ℝ² which is adapted to polar coordinates. We start by defining a product Fourier-Hankel transform f̂ and proving a sampling formula for f such that f̂ is compactly supported. For general f, the sampling formula and a partition of unity lead to an identity of the form $f = \sum_{μ, k, m} ⟨ f, φ_{μ k m} ⟩ ψ_{μ k m}$ , in which each function $φ_{μ k m}$ and $ψ_{μ k m}$ is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth...

The Lebesgue constants for the Franklin orthogonal system

Z. Ciesielski, A. Kamont (2004)

Studia Mathematica

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To each set of knots $t_{i} = i / 2 n$ for i = 0,...,2ν and $t_{i} = (i - ν) / n$ for i = 2ν + 1,..., n + ν, with 1 ≤ ν ≤ n, there corresponds the space $_{ν, n}$ of all piecewise linear and continuous functions on I = [0,1] with knots $t_{i}$ and the orthogonal projection $P_{ν, n}$ of L²(I) onto $_{ν, n}$ . The main result is $l i m_{(n - ν) \land ν \to \infty} | | P_{ν, n} | | ₁ = s u p_{ν, n : 1 \leq ν \leq n} | | P_{ν, n} | | ₁ = 2 + (2 - \sqrt 3) ²$ . This shows that the Lebesgue constant for the Franklin orthogonal system is 2 + (2-√3)².

Compactness criteria in function spaces

Monika Dörfler, Hans G. Feichtinger, Karlheinz Gröchenig (2002)

Colloquium Mathematicae

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The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for $L ² (ℝ^{d})$ , and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin, modulation and Bargmann-Fock spaces. ...

General Haar systems and greedy approximation

Anna Kamont (2001)

Studia Mathematica

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We show that each general Haar system is permutatively equivalent in $L^{p} ([0, 1])$ , 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^{p} ([0, 1])$ , 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^{p} ({[0, 1]}^{d})$ , 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases...

Function spaces with dominating mixed smoothness

Jan Vybiral

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We study several techniques which are well known in the case of Besov and Triebel-Lizorkin spaces and extend them to spaces with dominating mixed smoothness. We use the ideas of Triebel to prove three important decomposition theorems. We deal with so-called atomic, subatomic and wavelet decompositions. All these theorems have much in common. Roughly speaking, they say that a function f belongs to some function space (say $S_{p, q}^{r ̅} A$ ) if, and only if, it can be decomposed as $f (x) = \sum_{ν} \sum_{m} λ_{ν m} a_{ν m} (x)$ , convergence in S’, with...

Gabor meets Littlewood-Paley: Gabor expansions in $L^{p} (ℝ^{d})$

Karlheinz Gröchenig, Christopher Heil (2001)

Studia Mathematica

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It is known that Gabor expansions do not converge unconditionally in $L^{p}$ and that $L^{p}$ cannot be characterized in terms of the magnitudes of Gabor coefficients. By using a combination of Littlewood-Paley and Gabor theory, we show that $L^{p}$ can nevertheless be characterized in terms of Gabor expansions, and that the partial sums of Gabor expansions converge in $L^{p}$ -norm.

Pointwise regularity associated with function spaces and multifractal analysis

Stéphane Jaffard (2006)

Banach Center Publications

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The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces $C_{E}^{α} (x ₀)$ are constructed, leading to a notion of pointwise regularity with respect to E; the case $E = L^{\infty}$ corresponds to the usual Hölder regularity,...

Continuous rearrangements of the Haar system in $H_{p}$ for 0 < p < ∞

Krzysztof Smela (2008)

Studia Mathematica

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We prove three theorems on linear operators $T_{τ, p} : H_{p} (ℬ) \to H_{p}$ induced by rearrangement of a subsequence of a Haar system. We find a sufficient and necessary condition for $T_{τ, p}$ to be continuous for 0 < p < ∞.

Theoretical analysis for $ℓ_{1}$ - $ℓ_{2}$ minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

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We investigate the recovery of $k$ -sparse signals using the $ℓ_{1}$ - $ℓ_{2}$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$ -sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...

Continuous images of Lindelöf $p$ -groups, $σ$ -compact groups, and related results

Aleksander V. Arhangel'skii (2019)

Commentationes Mathematicae Universitatis Carolinae

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It is shown that there exists a $σ$ -compact topological group which cannot be represented as a continuous image of a Lindelöf $p$ -group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$ -groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$ -group, then there exists a covering $γ$ of $Y$ by dyadic compacta such that $| γ | \leq 2^{ω}$ . We also show that if a homogeneous compact space $Y$ is...

Localization techniques in $L^{p}$ spaces

A. Pełczyński, H. Rosenthal (1975)

Studia Mathematica

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Mobius invariant Besov spaces on the unit ball of $ℂ^{n}$

Małgorzata Michalska, Maria Nowak, Paweł Sobolewski (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We give new characterizations of the analytic Besov spaces $B_{p}$ on the unit ball $𝔹$ of $ℂ^{n}$ in terms of oscillations and integral means over some Euclidian balls contained in $𝔹$ .

On operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

A localization property for $B_{p q}^{s}$ and $F_{p q}^{s}$ spaces

Hans Triebel (1994)

Studia Mathematica

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Let $f^{j} = \sum_{k} a_{k} f (2^{j + 1} x - 2 k)$ , where the sum is taken over the lattice of all points k in $ℝ^{n}$ having integer-valued components, j∈ℕ and $a_{k} \in ℂ$ . Let $A_{p q}^{s}$ be either $B_{p q}^{s}$ or $F_{p q}^{s}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on $ℝ^{n} .$ The aim of the paper is to clarify under what conditions $∥ f^{j} | A_{p q}^{s} ∥$ is equivalent to $2^{j (s - n / p)} (\sum_{k} | a_{k} |^{p})^{1 / p} ∥ f | A_{p q}^{s} ∥$ .

A remark on extrapolation of rearrangement operators on dyadic $H^{s}$ , 0 < s ≤ 1

Stefan Geiss, Paul F. X. Müller, Veronika Pillwein (2005)

Studia Mathematica

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For an injective map τ acting on the dyadic subintervals of the unit interval [0,1) we define the rearrangement operator $T_{s}$ , 0 < s < 2, to be the linear extension of the map $(h_{I}) / {(| I |}^{1 / s}) \mapsto (h_{τ (I)}) (| τ (I) |^{1 / s})$ , where $h_{I}$ denotes the $L^{\infty}$ -normalized Haar function supported on the dyadic interval I. We prove the following extrapolation result: If there exists at least one 0 < s₀ < 2 such that $T_{s ₀}$ is bounded on $H^{s ₀}$ , then for all 0 < s < 2 the operator $T_{s}$ is bounded on $H^{s}$ .

$L_{p, q}$ spaces

Joseph Kupka

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CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p, q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p, q} (B)$ ........ 235. Examples in $L_{p, q}$ theory...................................................................................

Locally functionally countable subalgebra of $ℛ (L)$

M. Elyasi, A. A. Estaji, M. Robat Sarpoushi (2020)

Archivum Mathematicum

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Let $L_{c} (X) = {f \in C (X) : \bar{C_{f}} = X}$ , where $C_{f}$ is the union of all open subsets $U \subseteq X$ such that $| f (U) | \leq ℵ_{0}$ . In this paper, we present a pointfree topology version of $L_{c} (X)$ , named $ℛ_{ℓ c} (L)$ . We observe that $ℛ_{ℓ c} (L)$ enjoys most of the important properties shared by $ℛ (L)$ and $ℛ_{c} (L)$ , where $ℛ_{c} (L)$ is the pointfree version of all continuous functions of $C (X)$ with countable image. The interrelation between $ℛ (L)$ , $ℛ_{ℓ c} (L)$ , and $ℛ_{c} (L)$ is examined. We show that $L_{c} (X) ≅ ℛ_{ℓ c} (𝔒 (X))$ for any space $X$ . Frames $L$ for which $ℛ_{ℓ c} (L) = ℛ (L)$ are characterized.