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Daubechies wavelets on intervals with application to BVPs

Václav Finěk (2004)

Applications of Mathematics

In this paper, Daubechies wavelets on intervals are investigated. An analytic technique for evaluating various types of integrals containing the scaling functions is proposed; they are compared with classical techniques. Finally, these results are applied to two-point boundary value problems.

Decomposition systems for function spaces

G. Kyriazis (2003)

Studia Mathematica

Let Θ : = θ I e : e E , I D be a decomposition system for L ( d ) indexed over D, the set of dyadic cubes in d , and a finite set E, and let Θ ̃ : = Θ ̃ I e : e E , I D be the corresponding dual functionals. That is, for every f L ( d ) , f = e E I 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 systems...

Density estimation with quadratic loss: a confidence intervals method

Pierre Alquier (2008)

ESAIM: Probability and Statistics

We propose a feature selection method for density estimation with quadratic loss. This method relies on the study of unidimensional approximation models and on the definition of confidence regions for the density thanks to these models. It is quite general and includes cases of interest like detection of relevant wavelets coefficients or selection of support vectors in SVM. In the general case, we prove that every selected feature actually improves the performance of the estimator. In the case...

Dimension functions, scaling sequences, and wavelet sets

Arambašić Ljiljana, Damir Bakić, Rajna Rajić (2010)

Studia Mathematica

The paper is a continuation of our study of dimension functions of orthonormal wavelets on the real line with dyadic dilations. The main result of Section 2 is Theorem 2.8 which provides an explicit reconstruction of the underlying generalized multiresolution analysis for any MSF wavelet. In Section 3 we reobtain a result of Bownik, Rzeszotnik and Speegle which states that for each dimension function D there exists an MSF wavelet whose dimension function coincides with D. Our method provides a completely...

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