The Uncertainty Principle on Groups

Abstract

The classical uncertainty principle asserts that both a function and its Fourier transform cannot be largely concentrated on intervals of small measure. Donoho and Stark [SIAM J. App. Math., 49 (1989), pp. 906–931] have shown recently that both cannot be largely concentrated on any sets of small measure—in the case of functions on the line or functions on finite cyclic groups and with concentrations measured in $L^2 $. The purpose of this note is to extend these results to functions on $\sigma $-finite locally compact abelian groups, with concentrations measured in $L^p $, $1\leqq p\leqq 2$. The first uncertainty principle on groups, due to Matolcsi and Szucs [D. R. Acad. Sci. Paris, 277 (1973), pp. 841–843], deals with full (rather than large) concentration, asserting that if a function and its Fourier transform are supported by sets T and W, then the product of the Haar measures of T and W must be at least 1. For the case of full concentration in $R^n $, Benedicks [J. Math. Anal. Appl.,106 (1985), pp. 180–183] has shown that the product must be infinite, i.e., that full concentration is very different from large concentration.

Locations

• University of Minnesota Digital Conservancy (University of Minnesota) - View - PDF
• SIAM Journal on Applied Mathematics - View