2,657 research outputs found
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
Support theorem on R^n and non compact symmetric spaces
We consider convolution equations of the type f * T = g where f, g are in
L^p(R^n) and T is a compactly supported distribution. Under natural assumptions
on the zero set of the Fourier transform of T we show that f is compactly
supported, provided g is. Similar results are proved for non compact symmetric
spaces as well
Shearlets and Optimally Sparse Approximations
Multivariate functions are typically governed by anisotropic features such as
edges in images or shock fronts in solutions of transport-dominated equations.
One major goal both for the purpose of compression as well as for an efficient
analysis is the provision of optimally sparse approximations of such functions.
Recently, cartoon-like images were introduced in 2D and 3D as a suitable model
class, and approximation properties were measured by considering the decay rate
of the error of the best -term approximation. Shearlet systems are to
date the only representation system, which provide optimally sparse
approximations of this model class in 2D as well as 3D. Even more, in contrast
to all other directional representation systems, a theory for compactly
supported shearlet frames was derived which moreover also satisfy this
optimality benchmark. This chapter shall serve as an introduction to and a
survey about sparse approximations of cartoon-like images by band-limited and
also compactly supported shearlet frames as well as a reference for the
state-of-the-art of this research field.Comment: in "Shearlets: Multiscale Analysis for Multivariate Data",
Birkh\"auser-Springe
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