A complete analogue of Hardy's theorem on semisimple Lie groups

Rudra P. Sarkar¹

¹ Stat-Math Unit Indian Statistical Institute 203 B. T. Road Calcutta 700108, India

Colloquium Mathematicum, Tome 93 (2002) no. 1, pp. 27-40

Résumé

A result by G. H. Hardy ([11]) says that if $f$ and its Fourier transform $\widehat {f}$ are $O(|x|^m e^{-\alpha x^2})$ and $O(|x|^n e^{-x^2/{(4\alpha )}})$ respectively for some $m,n\ge 0$ and $\alpha >0$, then $f$ and $\widehat {f}$ are $P(x)e^{-\alpha x^2}$ and $P'(x)e^{-x^2/{(4\alpha )}}$ respectively for some polynomials $P$ and $P'$. If in particular $f$ is as above, but $\widehat {f}$ is $o(e^{-x^2/{(4\alpha )}})$, then $f= 0$. In this article we will prove a complete analogue of this result for connected noncompact semisimple Lie groups with finite center. Our proof can be carried over to the real reductive groups of the Harish-Chandra class.

Détail

DOI : 10.4064/cm93-1-4

Keywords: result hardy says its fourier transform widehat alpha x alpha respectively alpha widehat alpha x alpha respectively polynomials particular above widehat x alpha article prove complete analogue result connected noncompact semisimple lie groups finite center proof carried real reductive groups harish chandra class