Orbital decomposition and Fourier transform of invariant distributions on $SL(2, F)$

Distributions and Fourier Transform

Introduction. The theory of distributions, or generalized functions, provides a unified framework for performing standard calculus operations on nonsmooth functions, measures (such as the Dirac delta function), and even more general measure-like objects in the same way as they are done for smooth functions. In this theory, any distribution can be differentiated arbitrarily many times, a large c...

Explicit geodesic flow-invariant distributions using SL2(ℝ)-representation ladders

An explicit construction of a geodesic flow-invariant distribution lying in the discrete series of weight 2k isotopic component is found, using techniques from representation theory of SL2(R). It is found that the distribution represents an AC measure on the unit tangent bundle of the hyperbolic plane minus an explicit singular set. Finally, via an averaging argument, a geodesic flow-invariant ...

Fourier type transform on translation invariant valuations on convex sets

Let V be a finite dimensional real vector space. Let V alsm(V ) be the space of translation invariant smooth valuations on convex compact subsets of V . Let Dens(V ) be the space of Lebesgue measures on V . The goal of the article is to construct and study an isomorphism FV : V alsm(V )−̃→V alsm(V ∗)⊗Dens(V ) such that FV commutes with the natural action of the full linear group on both spaces, ...

Regularity of the Fourier transform on spaces of homogeneous distributions

1. Introduction. In this paper we consider the action of the Fourier transform on spaces of regular homogeneous distributions on R n , by which we mean to include the nonisotropic (sometimes called quasihomogeneous [6]) case. Specifically , consider the group of dilations of R n with positive scaling exponents dn x n) for δ > 0 (we will write this map as δ • x for short). Dilation induces scali...

12 Invariant Nonlinear Correlations via Fourier Transform