An uncertainty principle for ultraspherical expansions

[摘要] Motivated by Heisenberg-Weyl type uncertainty principles for the torus T and the sphere S-2 due to Breitenberger, Narowich, Ward, and others, we derive an uncertainty relation for radial functions on the spheres S-n subset of Rn+1 and, more generally, for ultraspherical expansions on [0, pi]. ln this setting, the ''frequency variance'' of a L-2-function on [0, pi] 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 [-pi, pi]. Moreover, using the densities f(t) of ''Gaussian measures'' on [0, pi] with the time t tending to 0, we show that the bound of our uncertainty principle is optimal. (C) 1997 Academic Press.

[发布日期] 1997-05-15