In this article we present a new approach to the computation of volume potentials over bounded domains, which is based on the quasi-interpolation of the density by smooth, almost locally supported basis functions for which the corresponding volume potentials are known. The quasi-interpolant is a linear combination of the basis function with shifted and scaled arguments and with coe cients expli...

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. For a given semi-rational polytope p and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope p parallel to the subspace L and sum...

There are three main problems for classical kernel density estimation in its application: boundary problem, over-smoothing problem of high (low)-density region and low-efficiency large samples. A new improved model multivariate adaptive binned quasi-interpolation based on a quadtree algorithm is proposed, which can avoid the deficiency improve precision model. The constructed steps. Firstly, th...

Our main concern in this paper is the design of simplified filtering procedures for the quasi-optimal approximation of functions in subspaces of L2 generated from the translates of a function φ(x). Examples of signal representations that fall into this framework are Schoenberg’s polynomial splines of degree n, and the various multiresolution spaces associated with the wavelet transform. After a...

We derive sharp and explicit upper bounds for possibly weighted Poincaré constants of finite element stars. The latter are star-shaped domains that consist of a finite number of nonoverlapping simplices or parallelepipeds which all share a common vertex. Bounds for Poincaré constants are needed in deriving error estimates for quasi-interpolation operators and a posteriori upper bounds.

In the study of approximation order, particularly in a multivariable setting, quasi-interpolants have played a major role. This report points out some limitations of quasi-interpolants and describes some recent results on approximation order obtained without the benefit of the quasi-interpolant idea. §1. Approximation Order In most general terms, “approximation order” is defined as follows. Def...

In this paper we present a new multilevel quasi-interpolation algorithm for smooth periodic functions using scaled Gaussians as basis functions. Recent research in this area has focussed upon implementations using basis function with finite smoothness. In this paper we deliver a first error estimates for the multilevel algorithm using analytic basis functions. The estimate has two parts, one in...

We present a general and simple procedure to construct quasi-interpolants in hierarchical spaces, which are composed of a hierarchy of nested spaces. The hierarchical quasi-interpolants are described in terms of the truncated hierarchical basis. Once for each level in the hierarchy a quasi-interpolant is selected in the corresponding space, the hierarchical quasi-interpolants are obtained witho...

Let s ≥ 1 be an integer, φ : Rs → R be a compactly supported function, and S(φ) denote the linear span of {φ(· − k) : k ∈ Zs}. We consider the problem of approximating a continuous function f : Rs → R on compact subsets of Rs from the classes S(φ(h·)), h > 0, based on samples of the function at scattered sites in R s. We demonstrate how classical polynomial inequalities lead to the construction...

We derive new convergence results for the Schoenberg operator and more general quasi-interpolation operators. In particular, we prove that natural conditions on the generator function imply convergence of these operators in the Fourier algebra A(Rd) = FL1(Rd) and in S0(R), a function space developed by the first author and often used in time-frequency analysis. As a simple yet very useful conse...