AN UNCERTAINTY PRINCIPLE RELATED TO THE POISSON SUMMATION FORMULA

We prove a class of uncertainty principles of the form \\S(g)f\\(1) le ss than or equal to C(\\x(alpha)f\\(p)+\\omega(b) (f) over cap\\(q)), where S(g)f is the short time Fourier transform of f. We obtain a char acterization of the range of parameters a, b, p, q for which such an u ncertainty principle holds. Counterexamples are constructed using Gabo r expansions and unimodular polynomials. These uncertainty principles relate the decay of f and (f) over cap to their behaviour in phase spa ce. Two applications are given: (a) If such an inequality holds, then the Poisson summation formula is valid with absolute convergence of bo th sums. (b) The validity of an uncertainty principle implies sufficie nt conditions on sigma symbol a such that the corresponding pseudodiff erential operator is of trace class.