Decomposition and reconstruction of multidimensional signals by radial functions with tension parameters

The aim of the paper is to construct a multiresolution analysis of L²(IR^d) based on generalized kernels which are fundamental solutions of differential operators of the form \(\boldsymbol {\prod }_{\ell =?1}^{m}(-{\Delta }+\kappa _{\ell }^{2}\,I)\). We study its properties and provide a set of pre-wavelets associated with it, as well as the filters which are indispensable to perform decomposition and reconstruction of a given signal, being very useful in applied problems thanks to the presence of the tension parameters κ_?.     

Irregularity detection from noisy data in one and two dimensions

In this work we consider the problem of detecting the irregularities of univariate functions from noisy data and its extension to bivariate functions which present lines of points of irregularity. This revised version was published online in August 2006 with corrections to the Cover Date.     

Generalized Whittle–Matérn and polyharmonic kernels

This paper simultaneously generalizes two standard classes of radial kernels, the polyharmonic kernels related to the differential operator (???Δ) ^m and the Whittle–Matérn kernels related to the differential operator (???Δ?+?I) ^m . This is done by allowing general differential operators of the form $\prod_{j=1}^m(-\Delta+\kappa_j^2I)$ with nonzero κ _j and calculating their associated kernels. It turns out that they can be explicity given by starting from scaled Whittle–Matérn kernels and taking divided differences with respect to their scale. They are positive definite radial kernels which are reproducing kernels in Hilbert spaces norm-equivalent to $W_2^m(\ensuremath{\mathbb{R}}^d)$ . On the side, we prove that generalized inverse multiquadric kernels of the form $\prod_{j=1}^m(r^2+\kappa_j^2)^{-1}$ are positive definite, and we provide their Fourier transforms. Surprisingly, these Fourier transforms lead to kernels of Whittle–Matérn form with a variable scale κ(r) between κ ₁,...,κ _m . We also consider the case where some of the κ _j vanish. This leads to conditionally positive definite kernels that are linear combinations of the above variable-scale Whittle–Matérn kernels and polyharmonic kernels. Some numerical examples are added for illustration.     

Polyharmonic multiresolution analysis: an overview and some new results

This paper first presents a condensed state of art on multiresolution analysis using polyharmonic splines: definition and main properties of polyharmonic splines, construction of B-splines and wavelets, decomposition and reconstruction filters; properties of the so-obtained operators, convergence result and applications are given. Second this paper presents some new results on this topic: scattered data wavelet, new polyharmonic scaling functions and associated filters. Fourier transform is of extensive use to derive the tools of the various multiresolution analysis.     

Radical trapping properties of 3-aryl-2H-benzo[1,4]oxazin-4-oxides

A series of 3-aryl-2H-benzo[1,4]oxazin-4-oxides was prepared, and their ability to trap free radicals was investigated by EPR spectroscopy. In organic solvents, these compounds were able to efficiently scavenge all carbon- and oxygen-centered radicals tested, giving very persistent aminoxyls, except with superoxide anion whose spin adducts were unstable. The main feature of these nitrones as spin traps lies in the possibility to recognize the initial radical trapped. In fact, besides a g-factor and aminoxyl nitrogen EPR coupling constant dependence on the species trapped, the EPR spectra also show different patterns due to hyperfine splittings characteristic of the radical scavenged. This last important feature was investigated by means of density functional theory calculations.