44 research outputs found
Non-local fractional derivatives. Discrete and continuous
We prove maximum and comparison principles for fractional discrete
derivatives in the integers. Regularity results when the space is a mesh of
length , and approximation theorems to the continuous fractional derivatives
are shown. When the functions are good enough, these approximation procedures
give a measure of the order of approximation. These results also allows us to
prove the coincidence, for good enough functions, of the Marchaud and
Gr\"unwald-Letnikov derivatives in every point and the speed of convergence to
the Gr\"unwald-Letnikov derivative. The fractional discrete derivative will be
also described as a Neumann-Dirichlet operator defined by a semi-discrete
extension problem. Some operators related to the Harmonic Analysis associated
to the discrete derivative will be also considered, in particular their
behavior in the Lebesgue spaces $\ell^p(\mathbb{Z}).
Discrete Hölder spaces and their characterization via semigroups associated with the discrete Laplacian and kernel estimates
In this paper, we characterize the discrete Hölder spaces by means of the heat and Poisson semigroups associated with the discrete Laplacian. These characterizations allow us to get regularity properties of fractional powers of the discrete Laplacian and the Bessel potentials along these spaces and also in the discrete Zygmund spaces in a more direct way than using the pointwise definition of the spaces. To obtain our results, it has been crucial to get boundedness properties of the heat and Poisson kernels and their derivatives in both space and time variables. We believe that these estimates are also of independent interest
The quasi-static plasmonic problem for polyhedra
We characterize the essential spectrum of the plasmonic problem for polyhedra
in . The description is particularly simple for convex polyhedra
and permittivities . The plasmonic problem is interpreted as a
spectral problem through a boundary integral operator, the direct value of the
double layer potential, also known as the Neumann--Poincar\'e operator. We
therefore study the spectral structure of the the double layer potential for
polyhedral cones and polyhedra.Comment: 31 pages, 2 figure
Non-local fractional derivatives. Discrete and continuous
We prove maximum and comparison principles for the discrete fractional derivatives in the integers. Regularity results when the space is a mesh of length h, and approximation theorems to the continuous fractional derivatives are shown. When the functions are good enough (Hölder continuous), these approximation procedures give a measure of the order of approximation. These results also allow us to prove the coincidence, for Hölder continuous functions, of the Marchaud and Grünwald–Letnikov derivatives in every point and the speed of convergence to the Grünwald–Letnikov derivative. The discrete fractional derivative will be also described as a Neumann–Dirichlet operator defined by a semi-discrete extension problem. Some operators related to the Harmonic Analysis associated to the discrete derivative will be also considered, in particular their behavior in the Lebesgue spaces lp(Z)
Variation operators for semigroups associated with Fourier-Bessel expansions
In this paper we establish -boundedness properties for variation
operators defined by semigroups associated with Fourier-Bessel expansions
The Hardy-Littlewood property and maximal operators associated with the inverse Gauss measure
In this paper we characterize the Banach lattices with the Hardy-Littlewood
property by using maximal operators defined by semigroups of operators
associated with the inverse Gauss measure.Comment: 37 page