27,873 research outputs found
The Takagi problem on the disk and bidisk
We give a new proof on the disk that a Pick problem can be solved by a
rational function that is unimodular on the unit circle and for which the
number of poles inside the disk is no more than the number of non-positive
eigenvalues of the Pick matrix. We use this method to find rational solutions
to Pick problems on the bidisk
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
Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
consisting of finitely or countably many distributional operators
, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function with respect to
now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator of is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant to data
values sampled from an unknown generalized Sobolev function at data sites
located in some set . We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator . Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D.
thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}
The Gaussian core model in high dimensions
We prove lower bounds for energy in the Gaussian core model, in which point
particles interact via a Gaussian potential. Under the potential function with , we show that no point
configuration in of density can have energy less than
as with and
fixed. This lower bound asymptotically matches the upper bound of obtained as the expectation in the Siegel mean value
theorem, and it is attained by random lattices. The proof is based on the
linear programming bound, and it uses an interpolation construction analogous
to those used for the Beurling-Selberg extremal problem in analytic number
theory. In the other direction, we prove that the upper bound of is no longer asymptotically sharp when . As
a consequence of our results, we obtain bounds in for the
minimal energy under inverse power laws with , and
these bounds are sharp to within a constant factor as with
fixed.Comment: 30 pages, 1 figur
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