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 and on ℝⁿ and on ⁿ.
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 and on ℝⁿ and on ⁿ.
G. Kyriazis (2003)
Studia Mathematica
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Let be a decomposition system for indexed over D, the set of dyadic cubes in , and a finite set E, and let be the corresponding dual functionals. That is, for every , . 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 , e ∈ E, I ∈ D. Typical examples of such decomposition...
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 and 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 is also shown.
Anna Kamont (2001)
Studia Mathematica
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We show that each general Haar system is permutatively equivalent in , 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 , 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each , 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...
Sarah V. Cook (2004)
Colloquium Mathematicae
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For a wavelet ψ of compact support, we define a square function and a maximal function NΛ. We then obtain the equivalence of these functions for 0 < p < ∞. We show this equivalence by using good-λ inequalities.
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 (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in (1 ≤ p < +∞).
Aydin Sh. Shukurov (2014)
Colloquium Mathematicae
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It is well known that if φ(t) ≡ t, then the system 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 is a basis in some Lebesgue space . The aim of this short note is to show that the answer to this question is negative.
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 , which is equivalent to the weight function being > 0 a.e.
Wolfgang Lusky (2003)
Studia Mathematica
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Let 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 -space, then both X and A have bases. We apply these results to show that the spaces and have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.
Dorothee D. Haroske, Leszek Skrzypczak (2013)
Studia Mathematica
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We study embeddings of spaces of Besov-Morrey type, , where is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of . This continues our earlier studies relating to the case of . Moreover, we also characterise embeddings into the scale of spaces or into the space of bounded continuous functions.
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 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 , which has been used in many areas of mathematics with important applications. The following continuous extension of the classical refinement equation has also various interesting...
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 , in which each function and is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth...
Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2018)
Commentationes Mathematicae Universitatis Carolinae
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Using the technique of Fraïssé theory, for every constant , we construct a universal object in the class of Banach spaces possessing a normalized -suppression unconditional Schauder basis.
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 is a smooth bounded domain, 1 ≤ p < ∞ and , then , 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 . The purpose of this paper is to obtain analogous asymptotic formulae...
Z. Ciesielski, A. Kamont (2004)
Studia Mathematica
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To each set of knots for i = 0,...,2ν and for i = 2ν + 1,..., n + ν, with 1 ≤ ν ≤ n, there corresponds the space of all piecewise linear and continuous functions on I = [0,1] with knots and the orthogonal projection of L²(I) onto . The main result is . This shows that the Lebesgue constant for the Franklin orthogonal system is 2 + (2-√3)².
Gideon Schechtman (2013)
Studia Mathematica
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If and 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 with norm embeds into L₁. This generalizes a recent result of Prochno and Schütt.
Dylan Airey, Bill Mance, Joseph Vandehey (2016)
Czechoslovak Mathematical Journal
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Let be a sequence of bases with . In the case when the are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose -Cantor series expansion is both -normal and -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of , and from this construction we can provide computable constructions of numbers with atypical normality properties. ...
Mohammed Sahmoudi, Mohammed Elhassani Charkani (2023)
Mathematica Bohemica
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Let be an extension of a number field , where satisfies the monic irreducible polynomial of prime degree belonging to ( is the ring of integers of ). The purpose of this paper is to study the monogenity of over by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field...
Min Tang, Yong-Gao Chen (2006)
Colloquium Mathematicae
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Let , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and for all n̅ ∈ Zₘ.
Taras O. Banakh, Joanna Garbulińska-Wegrzyn (2020)
Commentationes Mathematicae Universitatis Carolinae
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We observe that the notion of an almost -universal based Banach space, introduced in our earlier paper [1]: Banakh T., Garbulińska-Wegrzyn J., The universal Banach space with a -suppression unconditional basis, Comment. Math. Univ. Carolin. 59 (2018), no. 2, 195–206, is vacuous for . Taking into account this discovery, we reformulate Theorem 5.2 from [1] in order to guarantee that the main results of [1] remain valid.
Gevorkyan Gegham, Kamont Anna
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AbstractWe study general Franklin systems, i.e. systems of orthonormal piecewise linear functions corresponding to quasi-dyadic sequences of partitions of [0,1]. The following problems are treated: unconditionality of the general Franklin basis in , 1 < p < ∞, and , 1/2 < p ≤ 1; equivalent conditions for the unconditional convergence of the Franklin series in for 0< p ≤ 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces...
Daniel Skodlerack (2013)
Annales de l’institut Fourier
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Let be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let be the centralizer of a semisimple rational Lie algebra element of We prove that the Bruhat-Tits building of can be affinely and -equivariantly embedded in the Bruhat-Tits building of so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let and be maps from to which preserve the Moy–Prasad filtrations....
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 , 1 ≤ p < ∞. Although perhaps not probable, the latter...
Clemens Heuberger, Daniel Krenn (2013)
Journal de Théorie des Nombres de Bordeaux
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We consider digit expansions with an endomorphism of an Abelian group. In such a numeral system, the -NAF condition (each block of 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 admits an optimal -NAF). This result is then applied to imaginary quadratic bases, which are used for scalar...
Karlheinz Gröchenig, Christopher Heil (2001)
Studia Mathematica
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It is known that Gabor expansions do not converge unconditionally in and that cannot be characterized in terms of the magnitudes of Gabor coefficients. By using a combination of Littlewood-Paley and Gabor theory, we show that can nevertheless be characterized in terms of Gabor expansions, and that the partial sums of Gabor expansions converge in -norm.
Hamid Ben Yakkou, Jalal Didi (2024)
Mathematica Bohemica
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Let be a pure number field generated by a complex root of a monic irreducible polynomial , where , , are three positive natural integers. The purpose of this paper is to study the monogenity of . Our results are illustrated by some examples.