13,732 research outputs found
Laguerre semigroup and Dunkl operators
We construct a two-parameter family of actions \omega_{k,a} of the Lie
algebra sl(2,R) by differential-difference operators on R^N \setminus {0}.
Here, k is a multiplicity-function for the Dunkl operators, and a>0 arises from
the interpolation of the Weil representation of Mp(N,R) and the minimal unitary
representation of O(N+1,2) keeping smaller symmetries.
We prove that this action \omega_{k,a} lifts to a unitary representation of
the universal covering of SL(2,R), and can even be extended to a holomorphic
semigroup \Omega_{k,a}. In the k\equiv 0 case, our semigroup generalizes the
Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup by the
second author with G. Mano (a=1).
One boundary value of our semigroup \Omega_{k,a} provides us with
(k,a)-generalized Fourier transforms F_{k,a}, which includes the Dunkl
transform D_k (a=2) and a new unitary operator H_k (a=1), namely a Dunkl-Hankel
transform.
We establish the inversion formula, and a generalization of the Plancherel
theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty
inequality for F_{k,a}. We also find kernel functions for \Omega_{k,a} and
F_{k,a} for a=1,2 in terms of Bessel functions and the Dunkl intertwining
operator.Comment: final version (some few typos, updated references
On the computation of Gaussian quadrature rules for Chebyshev sets of linearly independent functions
We consider the computation of quadrature rules that are exact for a
Chebyshev set of linearly independent functions on an interval . A
general theory of Chebyshev sets guarantees the existence of rules with a
Gaussian property, in the sense that basis functions can be integrated
exactly with just points and weights. Moreover, all weights are positive
and the points lie inside the interval . However, the points are not the
roots of an orthogonal polynomial or any other known special function as in the
case of regular Gaussian quadrature. The rules are characterized by a nonlinear
system of equations, and earlier numerical methods have mostly focused on
finding suitable starting values for a Newton iteration to solve this system.
In this paper we describe an alternative scheme that is robust and generally
applicable for so-called complete Chebyshev sets. These are ordered Chebyshev
sets where the first elements also form a Chebyshev set for each . The
points of the quadrature rule are computed one by one, increasing exactness of
the rule in each step. Each step reduces to finding the unique root of a
univariate and monotonic function. As such, the scheme of this paper is
guaranteed to succeed. The quadrature rules are of interest for integrals with
non-smooth integrands that are not well approximated by polynomials
Stochastic Analysis of Gaussian Processes via Fredholm Representation
We show that every separable Gaussian process with integrable variance
function admits a Fredholm representation with respect to a Brownian motion. We
extend the Fredholm representation to a transfer principle and develop
stochastic analysis by using it. We show the convenience of the Fredholm
representation by giving applications to equivalence in law, bridges, series
expansions, stochastic differential equations and maximum likelihood
estimations
Tridiagonalization of the hypergeometric operator and the Racah-Wilson algebra
The algebraic underpinning of the tridiagonalization procedure is
investigated. The focus is put on the tridiagonalization of the hypergeometric
operator and its associated quadratic Jacobi algebra. It is shown that under
tridiagonalization, the quadratic Jacobi algebra becomes the quadratic
Racah-Wilson algebra associated to the generic Racah/Wilson polynomials. A
degenerate case leading to the Hahn algebra is also discussed.Comment: 14 pages; Section 3 revise
-Pseudo-Bosons, Complex Hermite Polynomials, and Integral Quantization
The -pseudo-boson formalism is illustrated with two examples.
The first one involves deformed complex Hermite polynomials built using
finite-dimensional irreducible representations of the group of invertible matrices with complex entries.
It reveals interesting aspects of these representations. The second example is
based on a pseudo-bosonic generalization of operator-valued functions of a
complex variable which resolves the identity. We show that such a
generalization allows one to obtain a quantum pseudo-bosonic version of the
complex plane viewed as the canonical phase space and to understand functions
of the pseudo-bosonic operators as the quantized versions of functions of a
complex variable
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