Local means and wavelets in function spaces

Hans Triebel

Displaying similar documents to “Local means and wavelets in function spaces”

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 < ∞.

Inequivalence of Wavelet Systems in $L ₁ (ℝ^{d})$ and $B V (ℝ^{d})$

Paweł Bechler (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Theorems stating sufficient conditions for the inequivalence of the d-variate Haar wavelet system and another wavelet system in the spaces $L ₁ (ℝ^{d})$ and $B V (ℝ^{d})$ are proved. These results are used to show that the Strömberg wavelet system and the system of continuous Daubechies wavelets with minimal supports are not equivalent to the Haar system in these spaces. A theorem stating that some systems of smooth Daubechies wavelets are not equivalent to the Haar system in $L ₁ (ℝ^{d})$ is also shown.

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.

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...

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...

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...

Unconditional biorthogonal wavelet bases in $L^{p} (ℝ^{d})$

Waldemar Pompe (2002)

Colloquium Mathematicae

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We prove that a biorthogonal wavelet basis yields an unconditional basis in all spaces $L^{p} (ℝ^{d})$ with 1 < p < ∞, provided the biorthogonal wavelet set functions satisfy weak decay conditions. The biorthogonal wavelet set is associated with an arbitrary dilation matrix in any dimension.

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...

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...

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. ...

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...

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.

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.

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,...

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 < +∞).

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.

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.

La Queste del Saint $G R_{a}$ (AL): a Computational Approach to Local Algebra

Teo Mora (1989)

Publications mathématiques et informatique de Rennes

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Weighted local Orlicz-Hardy spaces with applications to pseudo-differential operators

Dachun Yang, Sibei Yang

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Let Φ be a concave function on (0,∞) of strictly critical lower type index $p_{Φ} \in (0, 1]$ and $ω \in A_{\infty}^{l o c} (ℝ ⁿ)$ (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space $h_{ω}^{Φ} (ℝ ⁿ)$ via the local grand maximal function. Let $ρ (t) \equiv t^{- 1} / Φ^{- 1} (t^{- 1})$ for all t ∈ (0,∞). We also introduce the BMO-type space $b m o_{ρ, ω} (ℝ ⁿ)$ and establish the duality between $h_{ω}^{Φ} (ℝ ⁿ)$ and $b m o_{ρ, ω} (ℝ ⁿ)$ . Characterizations of $h_{ω}^{Φ} (ℝ ⁿ)$ , including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are...

Normal number constructions for Cantor series with slowly growing bases

Dylan Airey, Bill Mance, Joseph Vandehey (2016)

Czechoslovak Mathematical Journal

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Let $Q = {(q_{n})}_{n = 1}^{\infty}$ be a sequence of bases with $q_{i} \geq 2$ . In the case when the $q_{i}$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$ -Cantor series expansion is both $Q$ -normal and $Q$ -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$ , and from this construction we can provide computable constructions of numbers with atypical normality properties. ...

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 < ∞.

A regularity theory for scalar local minimizers of splitting-type variational integrals

Michael Bildhauer, Martin Fuchs, Xiao Zhong (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p, q)$ -growth with exponents $p \leq q < \infty$ and show for the scalar case that locally bounded local minimizers are of class $C^{1, μ}$ . Note that to our knowledge the only $C^{1, μ}$ -results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

On connectivity properties of hyperspaces of $R_{0}$ spaces

M. Mršević (1979)

Matematički Vesnik

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Towards the Jacquet conjecture on the Local Converse Problem for $p$ -adic ${GL}_{n}$

Dihua Jiang, Chufeng Nien, Shaun Stevens (2015)

Journal of the European Mathematical Society

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The Local Converse Problem is to determine how the family of the local gamma factors $γ (s, π \times τ, ψ)$ characterizes the isomorphism class of an irreducible admissible generic representation $π$ of ${GL}_{n} (F)$ , with $F$ a non-archimedean local field, where $τ$ runs through all irreducible supercuspidal representations of ${GL}_{r} (F)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r = 1, 2, ..., [\frac{n}{2}]$ . Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet...

Matrix subspaces of L₁

Gideon Schechtman (2013)

Studia Mathematica

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If $E = e_{i}$ and $F = f_{i}$ are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices $a_{i, j}$ with norm $| | a_{i, j} {| |}_{E (F)} = | | \sum_{k} | | \sum_{l} a_{k, l} f_{l} | | e_{k} | |$ embeds into L₁. This generalizes a recent result of Prochno and Schütt.

Optimality of the Width- $w$ Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Clemens Heuberger, Daniel Krenn (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider digit expansions $\sum_{j = 0}^{ℓ - 1} Φ^{j} (d_{j})$ with an endomorphism $Φ$ of an Abelian group. In such a numeral system, the $w$ -NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$ -NAF). This result is then applied to imaginary quadratic bases, which are used for scalar...

Remarks on $L B I$ -subalgebras of $C (X)$

Mehdi Parsinia (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $A (X)$ denote a subalgebra of $C (X)$ which is closed under local bounded inversion, briefly, an $L B I$ -subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151–163. By characterizing maximal ideals of $A (X)$ , we generalize the notion of $z_{A}^{β}$ -ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C (X)$ ...

Some weighted norm inequalities for a one-sided version of $g *_{λ}$

L. de Rosa, C. Segovia (2006)

Studia Mathematica

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We study the boundedness of the one-sided operator $g ⁺_{λ, φ}$ between the weighted spaces $L^{p} (M ¯ w)$ and $L^{p} (w)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g ⁺_{λ, φ}$ . For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g ⁺_{λ, φ}$ from $L^{p} ({(M ¯)}^{[p / 2] + 1} w)$ to $L^{p} (w)$ , where ${(M ¯)}^{k}$ denotes the operator M¯ iterated k times.

The centralizer of a classical group and Bruhat-Tits buildings

Daniel Skodlerack (2013)

Annales de l’institut Fourier

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Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G .$ We prove that the Bruhat-Tits building $𝔅^{1} (H)$ of $H$ can be affinely and $G$ -equivariantly embedded in the Bruhat-Tits building $𝔅^{1} (G)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^{'}$ be maps from $𝔅^{1} (H)$ to $𝔅^{1} (G)$ which preserve the Moy–Prasad filtrations....