Estimation of a smoothness parameter by spline wavelets

Magdalena Meller; Natalia Jarzębkowska

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Displaying similar documents to “Estimation of a smoothness parameter by spline wavelets”

A quadratic spline-wavelet basis on the interval

Černá, Dana, Finěk, Václav, Šimůnková, Martina

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In signal and image processing as well as in numerical solution of differential equations, wavelets with short support and with vanishing moments are important because they have good approximation properties and enable fast algorithms. A B-spline of order $m$ is a spline function that has minimal support among all compactly supported refinable functions with respect to a given smoothness. And recently, B. Han and Z. Shen constructed Riesz wavelet bases of $L_{2} (ℝ)$ with $m$ vanishing moments based...

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.

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...

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.

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...

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.

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.

Parallel implementation of Wavelet-Galerkin method

Finěk, Václav, Šimůnková, Martina

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We present here some details of our implementation of Wavelet-Galerkin method for Poisson equation in C language parallelized by POSIX threads library and show its performance in dimensions $d \in {3, 4, 5}$ .

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. ...

A note on the adaptive estimation of the differential entropy by wavelet methods

Christophe Chesneau, Fabien Navarro, Oana Silvia Serea (2017)

Commentationes Mathematicae Universitatis Carolinae

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In this note we consider the estimation of the differential entropy of a probability density function. We propose a new adaptive estimator based on a plug-in approach and wavelet methods. Under the mean $𝕃_{p}$ error, $p \geq 1$ , this estimator attains fast rates of convergence for a wide class of functions. We present simulation results in order to support our theoretical findings.

Weighted embedding theorems for radial Besov and Triebel-Lizorkin spaces

Pablo L. De Nápoli, Irene Drelichman, Nicolas Saintier (2016)

Studia Mathematica

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We study the continuity and compactness of embeddings for radial Besov and Triebel-Lizorkin spaces with weights in the Muckenhoupt class $A_{\infty}$ . The main tool is a discretization in terms of an almost orthogonal wavelet expansion adapted to the radial situation.

Unconditionality of orthogonal spline systems in $L^{p}$

Markus Passenbrunner (2014)

Studia Mathematica

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We prove that given any natural number k and any dense point sequence (tₙ), the corresponding orthonormal spline system is an unconditional basis in reflexive $L^{p}$ .

Solution of nonlinear Fredholm-Hammerstein integral equations by using semiorthogonal spline wavelets.

Lakestani, M., Razzaghi, M., Dehghan, M. (2005)

Mathematical Problems in Engineering

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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 < ∞.

Quantitative properties of quadratic spline wavelet bases in higher dimensions

Černá, Dana, Finěk, Václav, Šimůnková, Martina

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To use wavelets efficiently to solve numerically partial differential equations in higher dimensions, it is necessary to have at one’s disposal suitable wavelet bases. Ideal wavelets should have short supports and vanishing moments, be smooth and known in closed form, and a corresponding wavelet basis should be well-conditioned. In our contribution, we compare condition numbers of different quadratic spline wavelet bases in dimensions d = 1, 2 and 3 on tensor product domains (0,1)^d. ...

Application of cubic box spline wavelets in the analysis of signal singularities

Waldemar Rakowski (2015)

International Journal of Applied Mathematics and Computer Science

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In the subject literature, wavelets such as the Mexican hat (the second derivative of a Gaussian) or the quadratic box spline are commonly used for the task of singularity detection. The disadvantage of the Mexican hat, however, is its unlimited support; the disadvantage of the quadratic box spline is a phase shift introduced by the wavelet, making it difficult to locate singular points. The paper deals with the construction and properties of wavelets in the form of cubic box splines...

Semiorthogonal linear prewavelets on irregular meshes

Peter Oswald (2006)

Banach Center Publications

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We extend results on constructing semiorthogonal linear spline prewavelet systems in one and two dimensions to the case of irregular dyadic refinement. In the one-dimensional case, we obtain sharp two-sided inequalities for the $L_{p}$ -condition, 1 < p < ∞, of such systems.