This paper deals with density and regression estimation problems for functional data. Using wavelet bases for Hilbert spaces of functions, we develop a new adaptive procedure based on wavelet thresholding. We provide theoretical results on its asymptotic performances.

In this paper, we obtain symmetric C∞ real-valued tight wavelet frames in L2(R) with compact support and the spectral frame approximation order. Furthermore, we present a family of symmetric compactly supported C∞ orthonormal complex wavelets in L2(R). A complete analysis of nonstationary tight wavelet frames and orthonormal wavelet bases in L2(R) is given.

Most of the research work on wavelet analysis so far has been concentrated on wavelets on uniform meshes in Euclidean spaces. We are interested in wavelet bases for function spaces on bounded domains with possibly nonuniform or irregular meshes. For this purpose, we introduce the projection method for construction of wavelet bases. Let (Vn)n=0,1,2,... be a family of closed subspaces of a Hilber...

In this paper, we introduce the notion of biorthogonal wavelet packets associated with nonuniform multiresoltion analysis and study their characteristics by means of Fourier transform. Three biorthogonal formulas regarding these wavelet packets are established. Moreover, it is shown how to obtain several new Riesz bases of the space (R) by constructing a series of subspaces of these nonuniform ...

We study the multivariate approximation by certain partial sums (hyperbolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets. We characterize spaces of functions which have a prescribed approximation error by hyperbolic wavelet sums in terms of a K -functional and interpolation spaces. The results parallel those for hyperbolic trigonometric cross approximation of ...

In the present paper we find new constructions of orthonormal multiresolution analyses on the triangle ∆. In the first one, we describe a direct method to define an orthonormal multiresolution analysis R which is adapted for the study of the Sobolev spaces H 0(∆) (s ∈ N). In the second one, we add boundary conditions for constructing an orthonormal multiresolution analysis which is adapted for ...

We build wavelet-like functions based on a parametrized family of pseudo-differential operators L~ν that satisfy some admissibility and scalability conditions. The shifts of the generalized B-splines, which are localized versions of the Green function of L~ν , generate a family of L-spline spaces. These spaces have the approximation order equal to the order of the underlying operator. A sequenc...

We investigate the lifting scheme as a method for constructing compactly supported biorthogonal scaling functions and wavelets. A well-known issue arising with the use of this scheme is that the resulting functions are only formally biorthogonal. It is not guaranteed that the new wavelet bases actually exist in an acceptable sense. To verify that these bases do exist one must test an associated...

In this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be wr...

To give exibility to the time-frequency resolution trade-o of orthonormal (ON) wavelet bases constructed by I.Daubechies [2, 1], recently multiplicityM ON wavelet bases(more generally tight frames (TFs)) have been constructedby several authors [3, 4, ?, 7, 12]. These generalizations ofthe multiplicity 2 ON wavelet bases of I. Daubechies, di erfrom the latter in that, whi...