An Uncertainty Principle for Ultraspherical Expansions

Motivated by Heisenberg]Weyl type uncertainty principles for the torus T and the sphere S2 due to Breitenberger, Narowich, Ward, and others, we derive an uncertainty relation for radial functions on the spheres Sn ;Rnq1 and, more generally, for ultraspherical expansions onw0,px. In this setting, the ‘‘frequency variance’’ of a L2-function onw0,pxis 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 wyp,px. Moreover, using the densities ft of ‘‘Gaussian measures’’ onw0,pxwith the time t tending to 0, we show that the bound of our uncertainty principle is optimal.