Laguerre Polynomials, Restriction Principle, and Holomorphic Representations of SL(2,R)

The restriction principle is used to implement a realization of the holomorphic representations of SL(2,R) on L ² (R ⁺,t dt) by way of the standard upper half plane realization. The resulting unitary equivalence establishes a correspondence between functions that transform according to the character e^–i(2n++1); under rotations and the Laguerre polynomials. The standard recursion relations amongst Laguerre polynomials are derived from the action of the Lie algebra.     

Analytic continuation of Flensted-Jensen representations

LetX=G/H be an affine symmetric space of Hermitian type. For modules in the analytic continuation of the scalar holomorphic discrete series forG, we show existence and uniqueness (that is, a multiplicity one result) of imbeddings into functions onX. The corresponding intertwining operators are analyzed using our previous methods for discrete series. In a slightly less explicit way, we also give the analogous results for the continuation of the general discrete series.     

Holomorphic H-spherical distribution vectors in principal series representations

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L²(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown . In this article we construct a minimal G-invariant subdomain _H of with G/H as Shilov boundary. Let be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of , which admit a holomorphic extension to _H, is one dimensional. Furthermore we give a spectral definition of a Hardy space corresponding to those distribution vectors. In particular we achieve a geometric realization of a multiplicity free subspace of L²(G/H)_mc in a space of holomorphic functions.     

A Paley-Wiener theorem for the spherical Laplace transform on causal symmetric spaces of rank 1

We formulate and prove a topological Paley-Wiener theorem for the normalized spherical Laplace transform defined on the rank 1 causal symmetric spaces , for .

     

Support properties and Holmgren's uniqueness theorem for differential operators with hyperplane singularities

Let W be a finite Coxeter group acting linearly on Rn. In this article we study the support properties of a W-invariant partial differential operator D on Rn with real analytic coefficients. Our assumption is that the principal symbol of D has a special form, related to the root system corresponding to W. In particular the zeros of the principal symbol are supposed to be located on hyperplanes fixed by reflections in W. We show that conv(suppDf)=conv(suppf) holds for all compactly supported smooth functions f so that conv(suppf) is W-invariant. The main tools in the proof are Holmgren's uniqueness theorem and some elementary convex geometry. Several examples and applications linked to the theory of special functions associated with root systems are presented.     

Three-Way Tiling Sets in Two Dimensions

In this article we show that there exist measurable sets W⊂ℝ² with finite measure that tile ℝ² in a measurable way under the action of a expansive matrix A, an affine Weyl group [(W)\tilde]\widetilde{W} , and a full rank lattice [\varGamma\tilde] ì \mathbbR²\widetilde{\varGamma}\subset\mathbb{R}^{2} . This note is follow-up research to the earlier article “Coxeter groups and wavelet sets” by the first and second authors, and is also relevant to the earlier article “Coxeter groups, wavelets, multiresolution and sampling” by M. Dobrescu and the third author. After writing these two articles, the three authors participated in a workshop at the Banff Center on “Operator methods in fractal analysis, wavelets and dynamical systems,” December 2–7, 2006, organized by O. Bratteli, P. Jorgensen, D. Kribs, G. ólafsson, and S. Silvestrov, and discussed the interrelationships and differences between the articles, and worked on two open problems posed in the Larson-Massopust article. We solved part of Problem 2, including a surprising positive solution to a conjecture that was raised, and we present our results in this article.     

Fourier Series on Compact Symmetric Spaces: K-Finite Functions of Small Support

The Fourier coefficients of a function f on a compact symmetric space U/K are given by integration of f against matrix coefficients of irreducible representations of U. The coefficients depend on a spectral parameter μ, which determines the representation, and they can be represented by elements [^(f)](m)\hat{f}(\mu) in a common Hilbert space ℋ.     

Examples of Coorbit Spaces for Dual Pairs

In this paper we summarize and give examples of a generalization of the coorbit space theory initiated in the 1980’s by H.G. Feichtinger and K.H. Gröchenig. Coorbit theory has been a powerful tool in characterizing Banach spaces of distributions with the use of integrable representations of locally compact groups. Examples are a wavelet characterization of the Besov spaces and a characterization of some Bergman spaces by the discrete series representation of SL₂(?). We present examples of Banach spaces which could not be covered by the previous theory, and we also provide atomic decompositions for an example related to a non-integrable representation.