7,764 research outputs found
On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings
We study several fundamental harmonic analysis operators in the
multi-dimensional context of the Dunkl harmonic oscillator and the underlying
group of reflections isomorphic to . Noteworthy, we admit
negative values of the multiplicity functions. Our investigations include
maximal operators, -functions, Lusin area integrals, Riesz transforms and
multipliers of Laplace and Laplace-Stieltjes type. By means of the general
Calder\'on-Zygmund theory we prove that these operators are bounded on weighted
spaces, , and from weighted to weighted weak .
We also obtain similar results for analogous set of operators in the closely
related multi-dimensional Laguerre-symmetrized framework. The latter emerges
from a symmetrization procedure proposed recently by the first two authors. As
a by-product of the main developments we get some new results in the
multi-dimensional Laguerre function setting of convolution type
Sharp estimates of the spherical heat kernel
We prove sharp two-sided global estimates for the heat kernel associated with
a Euclidean sphere of arbitrary dimension. This solves a long-standing open
problem.Comment: 9 pages, to appear in J. Math. Pures Appl. (9
Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex
We prove genuinely sharp two-sided global estimates for heat kernels on all
compact rank-one symmetric spaces. This generalizes the authors' recent result
obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar
heat kernel bounds are shown in the context of classical Jacobi expansions, on
a ball and on a simplex. These results are more precise than the qualitatively
sharp Gaussian estimates proved recently by several authors.Comment: 16 page
New spectral relations between products and powers of isotropic random matrices
We show that the limiting eigenvalue density of the product of n identically
distributed random matrices from an isotropic unitary ensemble (IUE) is equal
to the eigenvalue density of n-th power of a single matrix from this ensemble,
in the limit when the size of the matrix tends to infinity. Using this
observation one can derive the limiting density of the product of n independent
identically distributed non-hermitian matrices with unitary invariant measures.
In this paper we discuss two examples: the product of n Girko-Ginibre matrices
and the product of n truncated unitary matrices. We also provide an evidence
that the result holds also for isotropic orthogonal ensembles (IOE).Comment: 8 pages, 3 figures (in version 2 we added a figure and discussion on
finite size effects for isotropic orthogonal ensemble
Riesz-Jacobi transforms as principal value integrals
We establish an integral representation for the Riesz transforms naturally
associated with classical Jacobi expansions. We prove that the Riesz-Jacobi
transforms of odd orders express as principal value integrals against kernels
having non-integrable singularities on the diagonal. On the other hand, we show
that the Riesz-Jacobi transforms of even orders are not singular operators. In
fact they are given as usual integrals against integrable kernels plus or
minus, depending on the order, the identity operator. Our analysis indicates
that similar results, existing in the literature and corresponding to several
other settings related to classical discrete and continuous orthogonal
expansions, should be reinvestigated so as to be refined and in some cases also
corrected.Comment: 30 page
Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices (The Extended Version)
We consider a product of an arbitrary number of independent rectangular
Gaussian random matrices. We derive the mean densities of its eigenvalues and
singular values in the thermodynamic limit, eventually verified numerically.
These densities are encoded in the form of the so called M-transforms, for
which polynomial equations are found. We exploit the methods of planar
diagrammatics, enhanced to the non-Hermitian case, and free random variables,
respectively; both are described in the appendices. As particular results of
these two main equations, we find the singular behavior of the spectral
densities near zero. Moreover, we propose a finite-size form of the spectral
density of the product close to the border of its eigenvalues' domain. Also,
led by the striking similarity between the two main equations, we put forward a
conjecture about a simple relationship between the eigenvalues and singular
values of any non-Hermitian random matrix whose spectrum exhibits rotational
symmetry around zero.Comment: 50 pages, 8 figures, to appear in the Proceedings of the 23rd Marian
Smoluchowski Symposium on Statistical Physics: "Random Matrices, Statistical
Physics and Information Theory," September 26-30, 2010, Krakow, Polan
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