Aline Bonami

Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms

We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on Rd which may be written as P(x)exp(-(Ax,x)), with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x)f(y). We then obtain similar results for the windowed Fourier transform (also known, up to elementary changes of functions, as the radar ambiguity function or the Wigner transform). We complete the paper with a sharp version of Heisenbergs inequality for this transform.

nth-order fractional Brownian motion and fractional Gaussian noises

A generalization of fractional Brownian motion (fBm) of parameter H in ]0, 1[ is proposed. More precisely, this work leads to nth-order fBm (n-fBm) of H parameter in ]n-1, n[, where n is any strictly positive integer. They include fBm for the special case n=1. Properties of these new processes are investigated. Their covariance function are given, and it is shown that they are self similar. In addition, their spectral shape is assessed as 1/f/sup /spl alpha// with /spl alpha/ belonging to ]1; +/spl infin/[, providing a larger framework than classical fBm. Special interest is given to their nth-order stationary increments, which extend fractional Gaussian noises. The covariance function and power spectral densities are calculated. The properties and signal processing tasks such as a Cholesky-type synthesis technique and a maximum likelihood estimation method of the H parameter are presented. The results show that the estimator is efficient (unbiased and reaches the Cramer-Rao lower bound) for a large majority of tested values.

Endpoint for the div-curl lemma in Hardy spaces

We give a div-curl type lemma for the wedge product of closed differential forms on Rn when they have coefficients respectively in a Hardy space and L∞ or BMO. In this latter case, the wedge product belongs to an appropriate Hardy-Orlicz space.

Band-limited wavelets

This manuscript represents a series of lectures, a “mini course.” presented by Guido Weiss during the Conference in Harmonic Analysis and Partial Differential Equations that was held in Miraflores de la Sierra in June 1992. This presentation can be regarded as an introduction to “wavelets” and this material is, for the most part, self-contained. The lectures, however, are aimed at an audience that knows the fundamentals of Harmonic Analysis. We are considering those wavelets whose Fourier transforms are compactly supported since many of their properties are more easily formulated and derived than those of the more general wavelets. As is pointed out in the text, however, much of the material presented here is true more generally. Much of this development appears, perhaps in different form, elsewhere; nevertheless, we do introduce some new results. All this is carefully explained in the text.

Boundedness of Bergman projections on tube domains over light cones

Abstract. Let

Uniform Approximation and Explicit Estimates for the Prolate Spheroidal Wave Functions

\Gamma

Analytic Besov spaces and Hardy-type inequalities in tube domains over symmetric cones

be the future light cone in

Wavelets obtained by continuous deformations of the Haar wavelet

{\Bbb R}^n

Spectral decay of time and frequency limiting operator

, and

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

\Omega={\Bbb R}^n +i\Gamma