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Representations of Jordan algebras and special functions

Giancarlo Travaglini — 1991

Colloquium Mathematicae

This paper is concerned with the action of a special formally real Jordan algebra U on an Euclidean space E, with the decomposition of E under this action and with an application of this decomposition to the study of Bessel functions on the self-adjoint homogeneous cone associated to U.

Polyhedral summability of multiple Fourier series (and explicit formulas for Dirichlet kernels on n and on compact Lie groups)

Giancarlo Travaglini — 1993

Colloquium Mathematicae

We study polyhedral Dirichlet kernels on the n-dimensional torus and we write a fairly simple formula which extends the one-dimensional identity j = - N N e i j t = s i n ( ( N + ( 1 / 2 ) ) t ) / s i n ( ( 1 / 2 ) t ) . We prove sharp results for the Lebesgue constants and for the pointwise boundedness of polyhedral Dirichlet kernels; we apply our results and methods to approximation theory, to more general summability methods and to Fourier series on compact Lie groups, where we write an asymptotic formula for the Dirichlet kernels.

Average decay of Fourier transforms and geometry of convex sets.

Luca BrandoliniMarco RigoliGiancarlo Travaglini — 1998

Revista Matemática Iberoamericana

Let B be a convex body in R, with piecewise smooth boundary and let ^χ denote the Fourier transform of its characteristic function. In this paper we determine the admissible decays of the spherical L averages of ^χ and we relate our analysis to a problem in the geometry of convex sets. As an application we obtain sharp results on the average number of integer lattice points in large bodies randomly positioned in the plane.

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