Introducing a suitable variational formulation for the local error of scattered data interpolation by radial basis functions OE(r), the error can be bounded by a term depending on the Fourier transform of the interpolated function f and a certain "Kriging function", which allows a formulation as an integral involving the Fourier transform of OE. The explicit construction of locally well--behaving admissible coefficient vectors makes the Kriging function bounded by some power of the local density h of data points. This leads to error estimates for interpolation of functions f whose Fourier transform f is "dominated" by the nonnegative Fourier transform / of /(x) = OE(kxk) in the sense R j f j 2 / \Gamma1 dt ! 1. Approximation orders are arbitrarily high for interpolation with Hardy multiquadrics, inverse multiquadrics and Gaussian kernels. This was also proven in recent papers by Madych and Nelson, using a reproducing kernel Hilbert space approach and requiring the same h...

We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...

We develop the theoretical foundation of a new representation of real numbers based on the infinite composition of linear fractional transformations (lft), equivalently the infiite product of matrices, with non-negative coefficients. Any rational interval in the one point compactification of the real line, represented by the unit circle S¹, is expressed as the image of the base interval [0�1] under an lft. A sequence of shrinking nested intervals is then represented by an infinite product of matrices with integer coefficients such that the first so-called sign matrix determines an interval on which the real number lies. The subsequent so-called digit matrices have non-negative integer coe cients and successively re ne that interval. Based on the classi cation of lft's according to their conjugacy classes and their geometric dynamics, we show that there is a canonical choice of four sign matrices which are generated by rotation of S¹ by =4. Furthermore, the ordinary signed digit representation of real numbers in a given base induces a canonical choice of digit matrices.

this paper, we prove that Ø c enjoys the following property: lim

The removal of the peculiar degeneration arising in the classical concepts of rest frame and time parameterization is at the heart of the recently formulated Equivalence Principle (EP). The latter, stating that all physical systems can be connected by a coordinate transformation to the free one with vanishing energy, univocally leads to the Quantum Stationary HJ Equation (QSHJE). This is a third–order non–linear differential equation which provides a trajectory representation of Quantum Mechanics (QM). The trajectories depend on the Planck length through hidden variables which arise as initial conditions. The formulation has manifest p–q duality, a consequence of the involutive nature of the Legendre transformation and of its recently observed relation with second–order linear differential equations. This reflects in an intrinsic ψ D –ψ duality between linearly independent solutions of the Schrödinger equation. Unlike Bohm’s theory, there is a non–trivial action even for bound states and no pilot–wave guide is present. A basic property of the formulation is that no use of any axiomatic interpretation of the wave–function is made. For example, tunnelling is a direct consequence of the quantum potential which differs from the Bohmian one and plays the role of particle’s self–energy. Furthermore, the QSHJE is defined only if the ratio ψ D /ψ is a local homeomorphism of the extended

For radial basis function interpolation of scattered data in IR d , the approximative reproduction of high-degree polynomials is studied. Results include uniform error bounds and convergence orders on compact sets. x1. Introduction We consider interpolation of real-valued functions f defined on a set \Omega ` IR d ; d 1. These functions are interpolated on a set X := fx 1 ; : : : ; xNX g of NX 1 pairwise distinct points x 1 ; : : : ; xNX in \Omega\Gamma Interpolation is done by linear combinations of translates \Phi(x \Gamma x j ) of a single continuous real-valued function \Phi defined on IR d . For various reasons it is sometimes necessary to add the space IP d m of d--variate polynomials of order not exceeding m to the interpolating functions. Interpolation is uniquely possible under the requirement If p 2 P d m satisfies p(x i ) = 0 for all x i 2 X then p = 0; (1) and if \Phi is conditionally positive definite of order m (see e.g. [8]): Definition 1. A function \Phi...

. This paper is about an inverse problem. We assume we are given a function f(x) which is some sum of ridge functions of the form P m i=1 g i (a i \Delta x) and we just know an upper bound on m. We seek to identify the functions g i and also the directions a i from such limited information. Several ways to solve this nonlinear problem are discussed in this work. x1. Introduction A ridge function is a multivariate function h : IR n ! IR of the simple form h(x 1 ; : : : ; xn ) = g(a 1 x 1 + \Delta \Delta \Delta + anxn ) = g(a \Delta x); where g : IR ! IR and a = (a 1 ; : : : ; an ) 2 IR n nf0g. In other words, it is a multivariate function constant on the parallel hyperplanes a \Delta x = c, c 2 IR. The vector a 2 IR n nf0g is generally called the direction. Ridge functions appear in various areas and under various guises. We find them in the area of partial differential equations (where they have been known for many, many years under the name of plane waves [7]). We al...

A group G is said to act partially on a set Y if there is a map : G ! I(Y ) into the semigroup of partial bijections on X such that (g)(h) (gh) and (g) = (g ) for all g; h 2 G. We prove that each partial group action is the restriction of a universal global group action, and show that this result provides the key to understanding Munn's proof of the P -theorem.

ABSTRACT. This paper develops some mollification formulas involving convolutions between popular radial basis function (RBF) basic functions Φ, and suitable mollifiers. Polyharmonic splines, scaled Bessel kernels (Matern functions) and compactly supported basic functions are considered. A typical result is that in R d the convolution of | • | β and ( • 2 + c 2) −(β+2d)/2 is the generalized multiquadric ( • 2 + c 2) β/2 up to a multiplicative constant. The constant depends on c> 0, β where ℜ(β)> −d, and d. An application which motivated the development of the formulas is a technique called implicit smoothing. This computationally efficient technique smooths a previously obtained RBF fit by replacing the basic function Φ with a smoother version Ψ during evaluation. 1.

We examine the proposal made recently that the su(3) modular invariant partition functions could be related to the geometry of the complex Fermat curves. Although a number of coincidences and similarities emerge between them and certain algebraic curves related to triangular billiards, their meaning remains obscure. In an attempt to go beyond the su(3) case, we show that any rational conformal field theory determines canonically a Riemann surface. a Member of the CNRS b Chercheur Qualifi'e FNRS 1 Introduction The partition function of a rational conformal field theory (RCFT) on a torus is subjected to modular invariance constraints. These constraints turn out to be very strong, and have led to the classification of families of models. The most celebrated achievement is the ADE classification of su(2) Wess--Zumino--Novikov--Witten models [1]. Its relationship with the classification of simply--laced Lie algebras, a one--to--one correspondence, is an a posteriori observation. Two v...