69,120 research outputs found
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}
Graph approximation and generalized Tikhonov regularization for signal deblurring
Given a compact linear operator \K, the (pseudo) inverse \K^\dagger is
usually substituted by a family of regularizing operators which
depends on \K itself. Naturally, in the actual computation we are forced to
approximate the true continuous operator \K with a discrete operator
\K^{(n)} characterized by a finesses discretization parameter , and
obtaining then a discretized family of regularizing operators
. In general, the numerical scheme applied to discretize \K
does not preserve, asymptotically, the full spectrum of \K. In the context of
a generalized Tikhonov-type regularization, we show that a graph-based
approximation scheme that guarantees, asymptotically, a zero maximum relative
spectral error can significantly improve the approximated solutions given by
. This approach is combined with a graph based regularization
technique with respect to the penalty term
Pseudoscalar Mesons in the SU(3) Linear Sigma Model with Gaussian Functional Approximation
We study the SU(3) linear sigma model for the pseudoscalar mesons in the
Gaussian Functional Approximation (GFA). We use the SU(3) linear sigma model
Lagrangian with nonet scalar and pseudo-scalar mesons including symmetry
breaking terms. In the GFA, we take the Gaussian Ansatz for the ground state
wave function and apply the variational method to minimize the ground state
energy. We derive the gap equations for the dressed meson masses, which are
actually just variational parameters in the GFA method. We use the
Bethe-Salpeter equation for meson-meson scattering which provides the masses of
the physical nonet mesons. We construct the projection operators for the flavor
SU(3) in order to work out the scattering T-matrix in an efficient way. In this
paper, we discuss the properties of the Nambu-Goldstone bosons in various
limits of the chiral symmetry.Comment: 28 pages, comments and suggestions welcom
On approximation of functions from Sobolev spaces on metric graphs
Some results on the approximation of functions from the Sobolev spaces on
metric graphs by step functions are obtained. The estimates are uniform with
respect to all graphs of a given finite length, and the constant factors in the
inequalities are sharp.Comment: 19 pages, 0 figure
Physical Aspects of Pseudo-Hermitian and -Symmetric Quantum Mechanics
For a non-Hermitian Hamiltonian H possessing a real spectrum, we introduce a
canonical orthonormal basis in which a previously introduced unitary mapping of
H to a Hermitian Hamiltonian h takes a simple form. We use this basis to
construct the observables O of the quantum mechanics based on H. In particular,
we introduce pseudo-Hermitian position and momentum operators and a
pseudo-Hermitian quantization scheme that relates the latter to the ordinary
classical position and momentum observables. These allow us to address the
problem of determining the conserved probability density and the underlying
classical system for pseudo-Hermitian and in particular PT-symmetric quantum
systems. As a concrete example we construct the Hermitian Hamiltonian h, the
physical observables O, the localized states, and the conserved probability
density for the non-Hermitian PT-symmetric square well. We achieve this by
employing an appropriate perturbation scheme. For this system, we conduct a
comprehensive study of both the kinematical and dynamical effects of the
non-Hermiticity of the Hamiltonian on various physical quantities. In
particular, we show that these effects are quantum mechanical in nature and
diminish in the classical limit. Our results provide an objective assessment of
the physical aspects of PT-symmetric quantum mechanics and clarify its
relationship with both the conventional quantum mechanics and the classical
mechanics.Comment: 45 pages, 13 figures, 2 table
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