157 research outputs found
Lattice points in polytopes, box splines, and Todd operators
Let be a list of vectors that is totally unimodular. In a previous
article the author proved that every real-valued function on the set of
interior lattice points of the zonotope defined by can be extended to a
function on the whole zonotope of the form in a unique way, where
is a differential operator that is contained in the so-called internal
\Pcal-space. In this paper we construct an explicit solution to this
interpolation problem in terms of Todd operators. As a corollary we obtain a
slight generalisation of the Khovanskii-Pukhlikov formula that relates the
volume and the number of integer points in a smooth lattice polytope.Comment: 15 pages, 4 figure
Monotonicity preserving approximation of multivariate scattered data
This paper describes a new method of monotone interpolation and smoothing of multivariate scattered data. It is based on the assumption that the function to be approximated is Lipschitz continuous. The method provides the optimal approximation in the worst case scenario and tight error bounds. Smoothing of noisy data subject to monotonicity constraints is converted into a quadratic programming problem. Estimation of the unknown Lipschitz constant from the data by sample splitting and cross-validation is described. Extension of the method for locally Lipschitz functions is presented.<br /
Approximate Approximations from scattered data
The aim of this paper is to extend the approximate quasi-interpolation on a
uniform grid by dilated shifts of a smooth and rapidly decaying function on a
uniform grid to scattered data quasi-interpolation. It is shown that high order
approximation of smooth functions up to some prescribed accuracy is possible,
if the basis functions, which are centered at the scattered nodes, are
multiplied by suitable polynomials such that their sum is an approximate
partition of unity. For Gaussian functions we propose a method to construct the
approximate partition of unity and describe the application of the new
quasi-interpolation approach to the cubature of multi-dimensional integral
operators.Comment: 29 pages, 17 figure
Recent progress on univariate and multivariate polynomial and spline quasi-interpolants
Polynomial and spline quasi-interpolants (QIs) are practical and effective approximation operators. Among their remarkable properties, let us cite for example: good shape properties, easy computation and evaluation (no linear system to solve), uniform boundedness independently of the degree (polynomials) or of the partition (splines), good approximation order. We shall emphasize new results on various types of univariate and multivariate polynomial or spline QIs, depending on the nature of coefficient functionals, which can be differential, discrete or integral. We shall also present some applications of QIs to numerical methods
Computation of Infinitesimals for a Group Action on a Multispace of One Independent Variable
This work expands upon the paper Geometric Foundations of Numerical
Algorithms and Symmetry by Peter Olver wherein Olver uses a moving frames
approach to examine the action of a group on a curve within a generalization of
jet space known as multispace. Here we seek to further study group actions on
the multispace of curves by computing the infinitesimals for a given action.
For the most part, we proceed formally, and produce in the multispace a
recursion relation that closely mimics the previously known prolongation
recursion relations for infinitesimals of a group action on jet space.Comment: 19 page
Discrete moving frames on lattice varieties and lattice based multispace
In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalization of the jet bundle that also generalizes Olver’s one dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame
The algebra of the box spline
In this paper we want to revisit results of Dahmen and Micchelli on
box-splines which we reinterpret and make more precise. We compare these ideas
with the work of Brion, Szenes, Vergne and others on polytopes and partition
functions.Comment: 69 page
On dimension and existence of local bases for multivariate spline spaces
AbstractWe consider spaces of splines in k variables of smoothness r and degree d defined on a polytope in Rk which has been divided into simplices. Bernstein-Bézier methods are used to develop a framework for analyzing dimension and basis questions. Dimension formulae and local bases are found for the case r = 0 and general k. The main result of the paper shows the existence of local bases for spaces of trivariate splines (where k = 3) whenever d > 8r
Box splines and the equivariant index theorem
In this article, we start to recall the inversion formula for the convolution
with the Box spline. The equivariant cohomology and the equivariant K-theory
with respect to a compact torus G of various spaces associated to a linear
action of G in a vector space M can be both described using some vector spaces
of distributions, on the dual of the group G or on the dual of its Lie algebra.
The morphism from K-theory to cohomology is analyzed and the multiplication by
the Todd class is shown to correspond to the operator (deconvolution) inverting
the semidiscrete convolution with a box spline. Finally, the multiplicities of
the index of a G-transversally elliptic operator on M are determined using the
infinitesimal index of the symbol.Comment: 44 page
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