An uncertainty principle for ultraspherical expansions

Abstract

Motivated by Heisenberg-Weyl type uncertainty principles for the torusTand the sphereS2due to Breitenberger, Narowich, Ward, and others, we derive an uncertainty relation for radial functions on the spheresSn⊂Rn+1and, more generally, for ultraspherical expansions on [0,π]. In this setting, the "frequency variance" of aL2-function on [0,π] is defined by means of the ultraspherical differential operator, which plays the role of a Laplacian. Our proof is based on a certain first-order differential-difference operator on the doubled interval [-π,π]. Moreover, using the densitiesftof "Gaussian measures" on [0,π] with the timettending to 0, we show that the bound of our uncertainty principle is optimal. © 1997 Academic Press.