Divided Differences

Starting with a novel definition of divided differences, this essay derives and discusses the basic properties of, and facts about, (univariate) divided differences.Comment: 24 page

Approximation orders of shift-invariant subspaces of W2s(Rd)W^s_2({\Bbb R}^d)

We extend the existing theory of approximation orders provided by shift-invariant subspaces of $L_2$ to the setting of Sobolev spaces, provide treatment of $L_2$ cases that have not been covered before, and apply our results to determine approximation order of solutions to a refinement equation with a higher-dimensional solution space.Comment: 49 page

Box spline prewavelets of small support

The purpose of this paper is the construction of bi- and trivariate prewavelets from box-spline spaces, \ie\ piecewise polynomials of fixed degree on a uniform mesh. They have especially small support and form Riesz bases of the wavelet spaces, so they are stable. In particular, the supports achieved are smaller than those of the prewavelets due to Riemenschneider and Shen in a recent, similar constructio

C2 piecewise cubic quasi-interpolants on a 6-direction mesh

We study two kinds of quasi-interpolants (abbr. QI) in the space of C2 piecewise cubics in the plane, or in a rectangular domain, endowed with the highly symmetric triangulation generated by a uniform 6-direction mesh. It has been proved recently that this space is generated by the integer translates of two multi-box splines. One kind of QIs is of differential type and the other of discrete type. As those QIs are exact on the space of cubic polynomials, their approximation order is 4 for sufficiently smooth functions. In addition, they exhibit nice superconvergent properties at some specific points. Moreover, the infinite norms of the discrete QIs being small, they give excellent approximations of a smooth function and of its first order partial derivatives. The approximation properties of the QIs are illustrated by numerical examples

Zonotopal algebra

A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This well-known line of study is particularly interesting in case n\eqbd\rank X \ll N. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\it external, central, and internal.} Each algebraic structure is given in terms of a pair of homogeneous polynomial ideals in $n$ variables that are dual to each other: one encodes properties of the arrangement ${\cal H}(X)$, while the other encodes by duality properties of the zonotope $Z(X)$. The algebraic structures are defined purely in terms of the combinatorial structure of $X$, but are subsequently proved to be equally obtainable by applying suitable algebro-analytic operations to either of $Z(X)$ or ${\cal H}(X)$. The theory is universal in the sense that it requires no assumptions on the map $X$ (the only exception being that the algebro-analytic operations on $Z(X)$ yield sought-for results only in case $X$ is unimodular), and provides new tools that can be used in enumerative combinatorics, graph theory, representation theory, polytope geometry, and approximation theory.Comment: 44 pages; updated to reflect referees' remarks and the developments in the area since the article first appeared on the arXi

The Stability of One-Step Schemes for First-Order Two-Point Boundary Value Problems

The stability of a finite difference scheme is related explicitly to the stability of the continuous problem being solved. At times, this gives materially better estimates for the stability constant than those obtained by the standard process of appealing to the stability of the numerical scheme for the associated initial value problem

Universal resonant ultracold molecular scattering

The elastic scattering amplitudes of indistinguishable, bosonic, strongly-polar molecules possess universal properties at the coldest temperatures due to wave propagation in the long-range dipole-dipole field. Universal scattering cross sections and anisotropic threshold angular distributions, independent of molecular species, result from careful tuning of the dipole moment with an applied electric field. Three distinct families of threshold resonances also occur for specific field strengths, and can be both qualitatively and quantitatively predicted using elementary adiabatic and semi-classical techniques. The temperatures and densities of heteronuclear molecular gases required to observe these univeral characteristics are predicted. PACS numbers: 34.50.Cx, 31.15.ap, 33.15.-e, 34.20.-bComment: 4 pages, 5 figure

Smoothing under Diffeomorphic Constraints with Homeomorphic Splines

In this paper we introduce a new class of diffeomorphic smoothers based on general spline smoothing techniques and on the use of some tools that have been recently developed in the context of image warping to compute smooth diffeomorphisms. This diffeomorphic spline is defined as the solution of an ordinary differential equation governed by an appropriate time-dependent vector field. This solution has a closed form expression which can be computed using classical unconstrained spline smoothing techniques. This method does not require the use of quadratic or linear programming under inequality constraints and has therefore a low computational cost. In a one dimensional setting incorporating diffeomorphic constraints is equivalent to impose monotonicity. Thus, as an illustration, it is shown that such a monotone spline can be used to monotonize any unconstrained estimator of a regression function, and that this monotone smoother inherits the convergence properties of the unconstrained estimator. Some numerical experiments are proposed to illustrate its finite sample performances, and to compare them with another monotone estimator. We also provide a two-dimensional application on the computation of diffeomorphisms for landmark and image matching

Convergence of abstract splines

AbstractShekhtman (J. Approx. Theory, 30(1980), 237–246) gives a sufficient condition for the convergence of abstract splines. We show that his condition is not necessary and give a related condition which is both necessary and sufficient. In the process, we also give a necessary and sufficient condition for a sequence of abstract spline projectors to be bounded