Hardy's theorem for the Jacobi transform
type: Item
instance of: scholarly article
title: Hardy's theorem for the Jacobi transform
zbMATH Open document ID: 1051.43003
author: Nils Byrial Andersen
published in: Hiroshima Mathematical Journal
publication date: 2004-05-27T00:00:00Z
review text: Let \(a, b, \lambda \in \mathbb{C}\), \(\rho = a + b + 1\), and let \(t\) be a strictly positive real variable. The Jacobi function \(\varphi_\lambda^{a,b}\) (of order \((a,b)\)) is defined by \[ \varphi_\lambda^{a,b}(t) = {}_2F_1 (\tfrac{1}{2}(\rho + i\lambda),\tfrac{1}{2}(\rho - i\lambda),a + 1;-\sinh^2(t)), \] where \({}_2F_1\) is the Gauss hypergeometric function, and the Jacobi transform (of order \((a,b)\)) is given by \[ \widehat{f}^{a,b}(\lambda) = \int_{\mathbb{R}_+}\,f(t) \varphi_\lambda^{a,b}(t)(2\sinh(t))^{2a+1}(2\cosh(t))^{2b+1}\,dt \] for all even functions \(f\) on \(\mathbb{R}\) and all complex numbers \(\lambda\) for which the right hand side is well-defined.\newline The following generalization of Hardy's theorem [\textit{G. H. Hardy}, J. Lond. Math. Soc. 8, 227--231 (1933; Zbl 0007.30403)] is proven in this paper. Let \(a,b \in \mathbb{C}\), \(a \not\in -\mathbb{N}\), and \(\rho = a+ b + 1\) . Assume that \(f\) is an even measurable function on \(\mathbb{R}\) satisfying \[ | f(t)| \leq C(1 + | t| )^Me^{-\Re\rho| t| }e^{-\alpha| t| ^2} \qquad (t \in \mathbb{R}) \] and \[ | \widehat{f}^{a,b}(\lambda)| \leq Ce^{-\beta| \lambda| ^2} \qquad(\lambda \in \mathbb{R}) \] for nonnegative real constants \(C, M, \alpha, \beta\) with \(M \geq \Re a + \frac{1}{2}\) and \(\alpha\beta = \frac{1}{4}\). Then there exists a constant \(c\) such that \(\widehat{f}^{a,b}(\lambda) = ce^{-\beta\lambda^2}\). This theorem is part of the main result of the paper: an \(L^p\) version of Hardy's theorem for the Jacobi transform, which is then applied to study the Fourier transform on real hyperbolic spaces. The author shows that Hardy's theorem can be generalized to functions on the real hyperbolic spaces \(\text{SO}_0(1,n)/ \text{SO}_0(n)\) \((n \in \mathbb{N}^*)\), where the subscript ``\(_0\)'' denotes the identity component, but not to the spaces \(\text{SO}_0(m,n)/ \text{SO}_0(m-1,n)\) with \(m,n \in \mathbb{N}^*\) and \(m > 1\).
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 2068258
zbMATH Keywords: Hardy's theorem
zbMATH Keywords: Jacobi transform
zbMATH Keywords: real hyperbolic space
zbMATH Keywords: Fourier transform
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1150997948
OpenAlex ID: W1587499143
DOI: 10.32917/HMJ/1150997948
description: scientific article; zbMATH DE number 2068258