8,696 research outputs found
On positive functions with positive Fourier transforms
Using the basis of Hermite-Fourier functions (i.e. the quantum oscillator
eigenstates) and the Sturm theorem, we derive the practical constraints for a
function and its Fourier transform to be both positive. We propose a
constructive method based on the algebra of Hermite polynomials. Applications
are extended to the 2-dimensional case (i.e. Fourier-Bessel transforms and the
algebra of Laguerre polynomials) and to adding constraints on derivatives, such
as monotonicity or convexity.Comment: 12 pages, 23 figures. High definition figures can be obtained upon
request at [email protected] or [email protected]
The expansion of some distributions into the Wiener series
The aim of this paper is to investigate a discrete integral transform on
the real line, which seems to be better adapted for some applications then
the Hermite transform (see for example [6]). Another complete orthonormal
system (CON) of functions on the real line, which was introduced by Wiener
is more appropriate for nonlinear differential equations of mathematical
physics
An Alternative Definition of the Hermite Polynomials Related to the Dunkl Laplacian
We introduce the so-called Clifford-Hermite polynomials in the framework of
Dunkl operators, based on the theory of Clifford analysis. Several properties
of these polynomials are obtained, such as a Rodrigues formula, a differential
equation and an explicit relation connecting them with the generalized Laguerre
polynomials. A link is established with the generalized Hermite polynomials
related to the Dunkl operators (see [R\"osler M., Comm. Math. Phys. 192 (1998),
519-542, q-alg/9703006]) as well as with the basis of the weighted
space introduced by Dunkl.Comment: This is a contribution to the Special Issue on Dunkl Operators and
Related Topics, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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