arXiv subjects
Publications and source records attributed to Gestur Olafsson.
Let X=G/K be a Riemannian symmetric space of the noncompact type. We give a short exposition of the representation theory related to X, and discuss its holomorphic extension to the complex crown, a G-invariant subdomain in the complexified symmetric space X_\C=G_\C/K_\C. Applications to the heat transform and the Radon transform for X are given.
The Fourier coefficients of a smooth $K$-invariant function on a compact symmetric space $M=U/K$ are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients
We develop integral geometry for non-compactly causal symmetric spaces. We define a complex horospherical transform and, for some cases, identify it with a Cauchy type integral.
We study the heat equation associated to a multiplicity function on a root system, where the corresponding Laplace operator has been defined by Heckman and Opdam. In particular, we describe the image of the associated Segal-Bargmann transform as a space of holomorphic functions. In the case where the multiplicity function corresponds to a Riemannian symmetric space G/K of noncompact type, we obtain a description of the image of the space of K-invariant L^2-function on G/K under the Segal-Bargmann transform associated to the heat equation on G/K, thus generalizing (and reproving) a result of B. Hall for spaces of complex type.
We address the function space theory associated with the Schroedinger operator H. The discussion is featured with the Poeschl-Teller potential in quantum physics. Using biorthogonal dyadic system, we introduce Besov spaces and Triebel-Lizorkin spaces (including Sobolev spaces) associated with H. We then use interpolation method to identify these spaces with the classical ones for a certain range of p,q> 1. A physical implication is that the corresponding wave function $ψ(t,x)=e^{-i t H }f(x)$ admits appropriate time decay in the Besov space scale.
Let $Ω$ be a symmetric cone and $V$ the corresponding simple Euclidean Jordan algebra. In \cite{ado,do,do04,doz2} we considered the family of generalized Laguerre functions on $Ω$ that generalize the classical Laguerre functions on $\mathbb{R}^+$. This family forms an orthogonal basis for the subspace of $L$-invariant functions in $L^2(Ω,dμ_ν)$, where $dμ_ν$ is a certain measure on the cone and where $L$ is the group of linear transformations on $V$ that leave the cone $Ω$ invariant and fix the identity in $Ω$. The space $L^2(Ω,dμ_ν)$ supports a highest weight representation of the group $G$ of holomorphic diffeomorphisms that act on the tube domain $T(Ω)=Ω+ iV.$ In this article we give an explicit formula for the action of the Lie algebra of $G$ and via this action determine second order differential operators which give differential recursion relations for the generalized Laguerre functions generalizing the classical creation, preservation, and annihilation relations for the Laguerre functions on $\mathbb{R}^+$.
This note is a continuation of the previous paper math.AP/0411383 by the same authors. Its purpose is to extend the results of math.AP/0411383 to the context of root systems with even multiplicities. Under the even multiplicity assumption, we prove a local Paley-Wiener theorem for the Jacobi transform and the strong Huygens' principle for the wave equation associated with the modified compact Laplace operator.
The heat kernel or Bargmann-Segal transform on a noncompact Riemannian symmetric space X=G/K maps a square integrable function on X to a holomorphic function on the complex crown. In this article we determine the range of this transform.
We define a complex horospherical transform for affine symmetric spaces of Hermitian type and show that it has no kernel on the holomorphic discrete series.
We prove a Paley-Wiener Theorem for a class of symmetric spaces of the compact type, in which all root multiplicities are even. This theorem characterizes functions of small support in terms of holomorphic extendability and exponential type of their (discrete) Fourier transforms. We also provide three independent new proofs of the strong Huygens' principle for a suitable constant shift of the wave equation on odd-dimensional spaces from our class.
Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L^2(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 $Ξ$. In this article we construct a minimal G-invariant subdomain $Ξ_H$ of $Ξ$ with G/H as Shilov boundary. Let $π$ be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of $π$, which admit a holomorphic extension to $Ξ_H$, 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 L^2(G/H)_mc in a space of holomorphic functions.
Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner-Pollacyck polynomials on the one side, and highest weight representations of Hermitian Lie groups on the other side. The representation theory is used to derive differential equations and recursion relations satisfied by those special functions.
Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. This article gives an overview over some well known results about the continuous and discrete wavelet transforms and groups acting on $\mathbb{R}^n$. We also show how this action can give rise to wavelets, and in particular, MSF wavelets)in $L^2(\mathbb{R}^n)$.
In our previous papers \cite{doz1,doz2} we studied Laguerre functions and polynomials on symmetric cones $Ω=H/L$. The Laguerre functions $\ell^ν_{\mathbf{n}}$, $\mathbf{n}\in\mathbfΛ$, form an orthogonal basis in $L^{2}(Ω,dμ_ν)^{L}$ and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations $(π_ν, \mathcal{H}_ν)$ of the automorphism group $G$ corresponding to a tube domain $T(Ω)$. In this article we consider the case where $Ω$ is the space of positive definite Hermitian matrices and $G=\mathrm{SU}(n,n)$. We describe the Lie algebraic realization of $π_ν$ acting in $L^{2}(Ω,dμ_ν)$ and use that to determine explicit differential equations and recurrence relations for the Laguerre functions.
Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. In this article we discuss the construction of frames for $L^2(\R^n)$ using the action of closed subgroups $H\subset \mathrm{GL}(n,\mathbb{R})$ such that $H$ has an open orbit $\cO$ in $\R^n$ under the action $(h,ω)\mapsto (h^{-1})^T(ω)$. If $H$ has the form $ANR$, where $A$ is simply connected and abelian, $N$ contains a co-compact discrete subgroup and $R$ is compact containing the stabilizer group of $ω\in\cO$ then we construct a frame for the space $L^2_{\cO}(\R^n)$ of $L^2$-functions whose Fourier transform is supported in $\cO$. We apply this to the case where $H^T=H$ and the stabilizer is a symmetric subgroup, a case discussed for the continuous wavelet transform in a paper by Fabec and Olafsson.
The $Θ$-spherical functions generalize the spherical functions on Riemannian symmetric spaces and the spherical functions on non-compactly causal symmetric spaces. In this article we consider the case of even multiplicity functions. We construct a differential shift operator $D_m$ with smooth coefficients which generates the $Θ$-spherical functions from finite sums of exponential functions. We then use this fact to prove a Paley-Wiener theorem for the $Θ$-spherical transfrom.
We give an overview on the c-function of a non-compactly causal symmetric space G/H and explain its interplay with harmonic analysis and representation theory.
Let $G/H$ be a semisimple symmetric space. Then the space $L^2(G/H)$ can be decomposed into a finite sum of series representations induced from parabolic subgroups of $G$. The most continuous part of the spectrum of $L^2(G/H)$ is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions living on a tube type bounded symmetric domain, containing $G/H$ as a boundary component. A boundary value map is constructed and we show that it induces an $G$-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part $L_{\rm mc}^2(G/H)$ of $L^2(G/H)$. We also relate our Hardy space with the classical Hardy space on the tube domain.