Pseudo-differential Operator Associated to the Radial Basis Functions under Tension

Radial basis functions under tension (RBFT) depend on a positive parameter, incorporate the concept of spline with tension and provide a convenient way for the control of the behavior of the interpolating surface. The RBFT involve a function which is not complicated than exponential and may be easily coded. In this paper, we show that the RBFT, as like thin plate spline, may be associated to a differential operator in a Beppo-Levi space type. Both smoothing and interpolating problems by RBFT are studied. Résumé. Les fonctions splines radiales sous tension dépendent d’un paramètre positif. Ces fonctions permettent d’incorporer un concept de tension pour toute dimension de l’espace. On montre dans ce papier que ces fonctions sont associées à un opérateur pseudo-différentiel dans un espace de type BeppoLevi et on étudie le problème d’interpolation et de lissage. Des examples numériques sont donnés pour illustrer ce type d’approximation. Introduction The radial basis functions under tension (RBFT) was introduced in [5]. The RBFT depend on a positive parameter and provide a convenient way of controlling the behavior of the interpolating surface. The aim of this paper is to give an explicit definition and construction of the associated native space of the RBFT, as a Beppo-Levi space type (see Theorem 2 and remark 2 in [5, p.141]). We show that the RBFT, as the thin plate splines, may be carried out from a (pseudo-) differential operator with a simple fundamental solution arising in the formulation of the RBFT. Furthermore, in this paper, we investigate both interpolating and smoothing problems. The spline under tension was first introduced by Schweikert [22] as a mean of eliminating extraneous inflection points in curve fitting by the cubic spline. With the proper use of the tension parameter, a user is able to reduce the length of the interpolating curve and hence remove extraneous bumps. In the limit, the curve reduces to piecewise linear interpolant as the tension parameter becomes large. The problem of interpolation by spline with the concept of tension has many applications, for instance, in geology for terrain modelling [15,17,23]. The thin plate splines under tension for two variables has been previously addressed by Franke and Nielson [11] and by Franke [12]. In [11] Franke and Nielson proposed a method of a construction of surfaces under tension based on a triangulation of the domain containing the scattered interpolating points. The second approach of Franke [12], for the construction of bivariate splines under tension, models a physical process of thin plate under tension. The approach of Franke [12], is approximately parallel that the development of Harder and Desmarais [14] for thin plate splines and based on experimental results. The basis function used by Franke [12], 1 L.M.P.A, Université du Littoral Côte d’Opale, 50 rue F. Buisson BP699, F-62228 Calais Cedex, France; e-mail: [email protected] c © EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:072007 ESAIM: PROCEEDINGS 73 resulted from the equation of thin plate subject to lateral loads and mid-plane forces which may be written in the general form as ∆W = 1 D (q + Nx ∂2W ∂x2 + Ny ∂2W ∂y2 − Nxy ∂ 2W ∂x∂y ) where W is the lateral deflection, q is the lateral load, Nx, Ny and Nxy are the mid-plane forces, and D depends on the properties of the plate material. In order to simplify the previous equation, Franke considered a thin plate under a point load at the origin and set Nx/D = Ny/D = τ and Nxy = 0, which gives an equation in the following form ∆W − τ∆W = p, where p is the point load at the origin, namely p(0) = q/d and p = 0 elsewhere. Radial symmetry is assumed and by setting V = ∆W = 1 r d dr (r dW dr ), one may obtain the polar form of the last equation d dr (r dV dr ) − τrV = 0. The last equation is a modified Bessel equation of order zero, which has for convenient solution with respect to the boundary conditions, the function −(2π)−1K0(τr), where K0 is a modified Bessel function of order zero. Finally, the basis function proposed by Franke [12] is Wτ (r) = −(2π)−1 ∫ r 0 t−1 ∫ t 0 K0(τs)dsdt + C, where C is a constant which may be equal to zero. In the terms of Franke [12], the lack of an elementary representation of the basis function is not a serious problem. Franke [12] proposed that the function can be approximated, either by numerical approximation of the integral formulation above, or by considering the function as solution of ordinary differential equation. By the assumption of point loads, the problem of interpolation of scattered data (xk, yk) ∈ R for k = 1, . . . , N by thin plate spline under tension, as proposed by Franke [12], has the following representation F (x, y) = ∑N k=1 AkWτ ( √ (x− xk) + (y − yk)) + a. A theoretical framework of the problem of thin plate splines with tension was presented in [2–4] in ddimensional space R. Let us now give a brief description of the spline under tension as was presented in [2–4]. Let X(R) be the space of distributions whose derivatives of order one and two are square integrable on R, X(R) = { u ∈ D′(Rd) : Du ∈ L(R) for |α| = 1, 2}, (1) where D′(Rd) is the space of distributions on R. In the space X(R), we consider the semi-scalar product (u|v)X = ∑