57,075 research outputs found
Approximation and interpolation employing divergence-free radial basis functions with applications
Approximation and interpolation employing radial basis functions has
found important applications since the early 1980's in areas such
as signal processing, medical imaging, as well as neural networks.
Several applications demand that certain physical properties be
fulfilled, such as a function being divergence free. No such class
of radial basis functions that reflects these physical properties
was known until 1994, when Narcowich and Ward introduced a family of
matrix-valued radial basis functions that are divergence free. They
also obtained error bounds and stability estimates for interpolation
by means of these functions. These divergence-free functions are
very smooth, and have unbounded support. In this thesis we
introduce a new class of matrix-valued radial basis functions that are
divergence free as well as compactly supported. This leads to the
possibility of applying fast solvers for inverting interpolation
matrices, as these matrices are not only symmetric and positive
definite, but also sparse because of this compact support. We develop
error bounds and stability estimates which hold for a broad class of
functions. We conclude with applications to the numerical solution of
the Navier-Stokes equation for certain incompressible fluid flows
Compactly supported radial basis functions: How and why?
Compactly supported basis functions are widely required and used in many applications. We explain why radial basis functions are preferred to multi-variate polynomials for scattered data approximation in high-dimensional space and give a brief description on how to construct the most commonly used compactly supported radial basis functions - the Wendland functions and the new found missing Wendland functions. One can construct a compactly supported radial basis function with required smoothness according to the procedure described here without sophisticated mathematics. Very short programs and extended tables for compactly supported radial basis functions are supplied
Toeplitz operators defined by sesquilinear forms: Fock space case
The classical theory of Toeplitz operators in spaces of analytic functions
deals usually with symbols that are bounded measurable functions on the domain
in question. A further extension of the theory was made for symbols being
unbounded functions, measures, and compactly supported distributions, all of
them subject to some restrictions.
In the context of a reproducing kernel Hilbert space we propose a certain
framework for a `maximally possible' extension of the notion of Toeplitz
operators for a `maximally wide' class of `highly singular' symbols. Using the
language of sesquilinear forms we describe a certain common pattern for a
variety of analytically defined forms which, besides covering all previously
considered cases, permits us to introduce a further substantial extension of a
class of admissible symbols that generate bounded Toeplitz operators.
Although our approach is unified for all reproducing kernel Hilbert spaces,
for concrete operator consideration in this paper we restrict ourselves to
Toeplitz operators acting on the standard Fock (or Segal-Bargmann) space
Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights
We prove in a direct fashion that a multidimensional probability measure is
determinate if the higher dimensional analogue of Carleman's condition is
satisfied. In that case, the polynomials, as well as certain proper subspaces
of the trigonometric functions, are dense in the associated L_p spaces for all
finite p. In particular these three statements hold if the reciprocal of a
quasi-analytic weight has finite integral under the measure. We give practical
examples of such weights, based on their classification.
As in the one dimensional case, the results on determinacy of measures
supported on R^n lead to sufficient conditions for determinacy of measures
supported in a positive convex cone, i.e. the higher dimensional analogue of
determinacy in the sense of Stieltjes.Comment: 20 pages, LaTeX 2e, no figures. Second and final version, with minor
corrections and an additional section on Stieltjes determinacy in arbitrary
dimension. Accepted by The Annals of Probabilit
Local interpolation schemes for landmark-based image registration: a comparison
In this paper we focus, from a mathematical point of view, on properties and
performances of some local interpolation schemes for landmark-based image
registration. Precisely, we consider modified Shepard's interpolants,
Wendland's functions, and Lobachevsky splines. They are quite unlike each
other, but all of them are compactly supported and enjoy interesting
theoretical and computational properties. In particular, we point out some
unusual forms of the considered functions. Finally, detailed numerical
comparisons are given, considering also Gaussians and thin plate splines, which
are really globally supported but widely used in applications
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