Partitions of Unity and Approximation
C. de Boor,Ronald A. DeVore +1 more
- Vol. 93, Iss: 4, pp 705-709
TLDR
For translation invariant spaces S, a necessary and sufficient condition for the eventual denseness of the corresponding scaled spaces S sub h is that S contain a stable and locally supported partition of unity as discussed by the authors.Abstract:
: This document shows that for certain translation invariant spaces S, a necessary and sufficient condition for the eventual denseness of the corresponding scaled spaces S sub h is that S contain a stable and locally supported partition of unity. These results have been motivated by recent work on approximation by multivariate piecewise polynomials on regular meshes. (Author)read more
Citations
More filters
Journal ArticleDOI
Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces
TL;DR: A unified framework for uniform and nonuniform sampling and reconstruction in shift-invariant subspaces is provided by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling.
Journal ArticleDOI
The structure of finitely generated shift-invariant spaces in L2 (Rd)
TL;DR: In this paper, a simple characterization is given of finitely generated subspaces of L2(Rd) which are invariant under translation by any (multi)integer, and is used to give conditions under which such a space has a particularly nice generating set, namely a basis, and more than that, a basis with desirable properties, such as stability, orthogonality, or linear independence.
Book ChapterDOI
Quasiinterpolants and Approximation Power of Multivariate Splines
TL;DR: In this paper, the determination of the approximation power of spaces of multivariate splines with the aid of quasiinterpolants is reviewed and a streamlined description of the existing quasi-interpolant theory is given.
Journal ArticleDOI
Shift-invariant spaces on the real line
TL;DR: In this paper, the authors investigated the structure of shift-invariant spaces generated by a finite number of compactly supported functions in Lp(R) (1 ≤ p ≤ ∞) based on a study of linear independence of the shifts of the generators.
Patent
System and methods of nonuniform data sampling and data reconstruction in shift invariant and wavelet spaces
TL;DR: In this article, the authors propose a system and methods for converting data for an object of interest, which is characterized by a function between a digital form and an analog form, and reconstructing the object from the new data set.
References
More filters
Journal ArticleDOI
Approximation by smooth multivariate splines
C. de Boor,Ronald A. DeVore +1 more
TL;DR: In this article, the authors showed that piecewise polynomials of total degree r on a rectangular grid with all derivatives of order or = RHo continuous will not approximate certain smooth functions at all unless RHo is kept below (r-3)/2.
Journal ArticleDOI
Bivariate box splines and smooth pp functions on a three direction mesh
C. de Boor,Klaus Höllig +1 more
TL;DR: In this article, the authors construct a basis for S in terms of box splines and truncated powers, which allows them to determine the polynomials which are locally contained in S and to give upper and lower bounds for the degree of approximation.
Journal ArticleDOI
Approximation order from bivariate ¹-cubics: a counterexample
C. de Boor,Klaus Höllig +1 more
TL;DR: It is shown that the space of bivariate C1 piecewise cubic functions on a hexagonal mesh of size h approximates to certain smooth functions only to within O(h3) even though it contains a local partition of every cubic polynomial.
Journal ArticleDOI
On the optimal approximation rates for criss-cross finite element spaces
TL;DR: In this paper, it was shown that the approximation properties of l k, h μ are completely governed by those of the space spanned by the translates of all so-called box splines contained in L k,h μ.
Approximation by Smooth Bivariate Splines on a Three-Direction Mesh.
TL;DR: A basic question is to ascertain, for a given mesh delta and a family S of splines on delta, what its optimal approximation order is.
Related Papers (5)
The structure of finitely generated shift-invariant spaces in L2 (Rd)
Sampling procedures in function spaces and asymptotic equivalence with shannon's sampling theory
Akram Aldroubi,Michael Unser +1 more