Uncertainty principles for compact groups

Date of Completion

January 2008






Uncertainty principles assert, roughly, that a function and its Fourier transform cannot simultaneously be highly concentrated. Several uncertainty principles have been formulated for complex-valued functions on groups. For finite abelian groups, perhaps the most basic of these is an inequality which relates the sizes of the supports of f and its transform to the size of the group. In this work, we extend several previously known uncertainty principles for groups; we formulate a general operator-theoretic uncertainty principle for certain bounded operators on L2(G), for G an arbitrary compact groups. Our principle implies that an arbitrary nonzero function in L2(G) satisfies suppf r∈G&d4; dr rkf&d4; r≥1, where :·: denotes normalized Haar measure. For finite G, our principle has a nice operator-theoretic corollary. It states that if P and R are projection operators on the group algebra C G, such that P commutes with projection onto elements of G, and R commutes with left-multiplication, then &dvbm0;PR&dvbm0;2rkP˙ rkRG . The aforementioned corollaries extend several previous results, which we discuss in detail. We also provide alternative proofs of our results in the setting of finite groups, using only basic results from representation theory. ^