Christophe Rabut

Elementary m-harmonic cardinal B-splines

We generalize the notion of B-spline to the thin plate splines and to otherd-dimensional polyharmonic splines as defined in [Duchon, [3]]; for regular nets, we give the main properties of these “B-splines”: Fourier transform, decay when ∥x∥ → ∞, stability, integration property, links between B-splines of different orders or of different dimensions and in particular link with the polynomial B-splines, approximation using B-splines... We show that, in some sense, B-splines may be considered as a regularized form of the Dirac distribution.

Using the refinement equation for the construction of pre-wavelets III: Elliptic splines

The purpose of this paper is to provide multiresolution analysis, stationary subdivision and pre-wavelet decomposition onL2(Rd) based on a general class of functions which includes polyharmonic B-splines.

Vectorial interpolation using radial-basis-like functions

This paper deals with vector field interpolation, i.e., the data are R3 values located in scattered R3 points, while the interpolating function is a function from R3 into R3. In order to take into account possible connections between the components of the interpolant, we derive it by solving a variational spline problem involving the rotational and the divergence of the interpolant, and depending on a parameter ρ significative of the balance of the rotational part and of the divergence part, and on the order m of derivatives of the rotational and divergence involved in the minimized seminorm. We so obtain interpolants whose expression is σ(x) = Σni=1 Φ(x − xi)ai + pm−1(x), where Φ is some 3 × 3 matrix function, pm−1 is a degree m − 1 vectorial polynomial, and where the ai are R3-vectors. Besides, the ai meet a relation generalizing the usual orthogonality to all polynomials of degree at most m − 1. For ρ = 1, we find the usual m-harmonic splines in each component of σ. Numerical examples show the interest of the method, and we compare the so-obtained functions with the ones obtained by Matlabs procedures.

High levelm-harmonic cardinal B-splines

We generalize the notion ofm-harmonic cardinal B-spline defined in [Rabut, [6c]] to obtain “B-splines” on an infinite regular grid, which are halfway between “elementary B-splines” and the Lagrangean cardinal spline function. We give the main properties of these functions: Fourier transform, decay when ∥x∥ → ∞, integration,Pk-reproduction (fork<-2m−1) of the associated B-spline approximation, etc. We show that, in some sense, “high levelm-harmonic B-splines” may be considered as a finer regular approximation of the Dirac distribution than the elementarym-harmonic B-splines are.

AN INTRODUCTION TO SCHOENBERG'S APPROXIMATION

For a given function B and a non-zero real number h, Schoenbergs approximation defines from some data (jh, yj)jϵZd the function ΣjϵZd yj B(•h − j). For people not used to this kind of approximation, this paper intends to do a summary of the main definitions, properties and utilizations of Schoenbergs approximation: we show that the main tool to handle Schoenbergs approximation is the Fourier transform of B and even more its modified version, the transfer function of B; we give conditions for convergence of ΣjϵZd f(jh) B(•h − j) when h tends to zero, and we give various ways to define various B as combinations of translates of some function ϕ (usually ϕ is either some radial function, or obtained by a tensor product of some radial function), depending on the properties we want for the associated Schoenbergs approximation. Last, we show how multi-resolution analysis, subdivision techniques, and wavelets techniques, are nicely connected to Schoenbergs approximation.

Irrotational or divergence-free interpolation

Summary.This paper deals with interpolation of vectorial data in We build specific vectorial splines which interpolate the data and meet some required vectorial properties. We so derive irrotational or divergence-free interpolating splines. Numerical examples compare the so-obtained interpolating functions with the ones obtained by independent interpolation on each variable.

Multivariate Divided Differences with Simple Knots

In d dimensions (

Locally tensor product functions

d\geq1

On Pierre Bézier's life and motivations

), the order

How to build quasi-interpolants: application to polyharmonic B-Splines

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