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564
to Box Spline Theory
"... We investigate the relation between an ideal I of finite codimension in the space Π of multivariate polynomials and ideals which are generated by lower order perturbations of some generators for I. Of particular intereest are the codimension of these ideals and the local approximation order of their ..."
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of their kernels. The discussion, stimulated by recent results in approximation theory, allows us to provide a simple analysis of the polynomial and exponential spaces associated with box splines. We describe their structure, dimension, local approximation order and an algorithm for their construction
Ideal spatial adaptation by wavelet shrinkage
 Biometrika
, 1994
"... With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle o ers dramatic ad ..."
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Cited by 1269 (5 self)
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With ideal spatial adaptation, an oracle furnishes information about how best to adapt a spatially variable estimator, whether piecewise constant, piecewise polynomial, variable knot spline, or variable bandwidth kernel, to the unknown function. Estimation with the aid of an oracle o ers dramatic
Regularization Theory and Neural Networks Architectures
 Neural Computation
, 1995
"... We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Ba ..."
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Cited by 395 (32 self)
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to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models
On Polynomial Ideals of Finite Codimension with Applications to Box Spline Theory
 J. Math. Anal. Appl
, 1991
"... We investigate the relation between an ideal I of finite codimension in the space # of multivariate polynomials and ideals which are generated by lower order perturbations of some generators for I. Of particular intereest are the codimension of these ideals and the local approximation order of their ..."
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Cited by 26 (10 self)
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of their kernels. The discussion, stimulated by recent results in approximation theory, allows us to provide a simple analysis of the polynomial and exponential spaces associated with box splines. We describe their structure, dimension, local approximation order and an algorithm for their construction
The QuasiInterpolant as a Tool in Elementary Polynomial Spline Theory
, 1973
"... ll of the t j 's coincide. I leave unresolved any possible ambiguity when t = t j for some j, and concern myself only with left and right limits at such a point; i.e., I replace each t = t j by the "two points" t  j and t + j . As is well known, N ik > 0 on (t i , t i+k ), an ..."
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Cited by 27 (9 self)
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ll of the t j 's coincide. I leave unresolved any possible ambiguity when t = t j for some j, and concern myself only with left and right limits at such a point; i.e., I replace each t = t j by the "two points" t  j and t + j . As is well known, N ik > 0 on (t i , t i+k ), and N ik = 0 o# [t + i , t  i+k ] so that (since t i < t i+k , by assumption) N ik is not identically zero, while on the other hand, no more than k of the N jk 's are nonzero at any particular point. Consequently, for an arbitrary a #<F1
On the evaluation of box splines
"... The first (and for some still the only) multivariate Bspline is what today one would call the simplex spline, since it is derived from a simplex, and in distinction to other polyhedral splines, such as the cone spline and the box spline. The simplex spline was first talked about in 1976. However, i ..."
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Cited by 201 (8 self)
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. This is not the place to give an introduction to box spline theory, nor to review the relevant literature. Rather, the reader is urged to consult [de Boor, Hollig, Riemenschneider, 1993] for missing details and the proper literature references (as well as for many illustrations, two of which are reproduced below).
Polynomial Splines and Their Tensor Products in Extended Linear Modeling
 Ann. Statist
, 1997
"... ANOVA type models are considered for a regression function or for the logarithm of a probability function, conditional probability function, density function, conditional density function, hazard function, conditional hazard function, or spectral density function. Polynomial splines are used to m ..."
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Cited by 221 (16 self)
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ANOVA type models are considered for a regression function or for the logarithm of a probability function, conditional probability function, density function, conditional density function, hazard function, conditional hazard function, or spectral density function. Polynomial splines are used
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 339 (27 self)
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to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation
CENTER FOR THE MATHEMATICAL SCIENCES On Polynomial Ideals of Finite Codimension with Applications to Box Spline Theory
, 1988
"... AVeiftvestigatehere the relations between an ideal I of finite codimension in the space of multivariate polynomials and various ideals which are generated by lower order perturbations of the generators of I. Special emphasis is given to the question of the codimension of I and its perturbed counter ..."
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counterpart and to the local approximation order of their kernels. The discussion, stimulated by certain results in approximation theory, allows us to provide a simple analysis of the polynomial and exponential spaces associated with box splines. This includes their structure, dimension, local approximation
Results 1  10
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564