3,719 research outputs found
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
We extend the existing theory of approximation orders provided by
shift-invariant subspaces of to the setting of Sobolev spaces, provide
treatment of 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 of is captured in the study of its associated zonotope
, and, by duality, its associated hyperplane arrangement .
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 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 variables that are dual to each other: one encodes
properties of the arrangement , while the other encodes by duality
properties of the zonotope . The algebraic structures are defined purely
in terms of the combinatorial structure of , but are subsequently proved to
be equally obtainable by applying suitable algebro-analytic operations to
either of or . The theory is universal in the sense that it
requires no assumptions on the map (the only exception being that the
algebro-analytic operations on yield sought-for results only in case
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
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