We construct the Haar wavelets on a local field K of positive characteristic and show that the Haar wavelet system forms an unconditional basis for $L^p\left(K\right)$, 1 < p < ∞. We also prove that this system, normalized in $L^p\left(K\right)$, is a democratic basis of $L^p\left(K\right)$. This also proves that the Haar system is a greedy basis of $L^p\left(K\right)$ for 1 < p < ∞.

This paper describes expansions in terms of Hermite and Laguerre functions similar to the Frazier-Jawerth expansion in Fourier analysis. The wave packets occurring in these expansions are finite linear combinations of Hermite and Laguerre functions. The Shannon sampling formula played an important role in the derivation of the Frazier-Jawerth expansion. In this paper we use the Christoffel-Darboux formula for orthogonal polynomials instead. We obtain estimates on the decay of the Hermite and Laguerre...

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...