On the uncertainty relation for positive-definite probability densities, II

Let P denote the class of probability density functions on ℝ with a nonnegative and integrable characteristic function. To each p ε P, there is an adjoint density p̂, which is proportional to the characteristic function of p. The products λ(p) = Var(p) Var(p̂) have a greatest lower bound Λ, and it is known that 0.5276 < Λ < 0.8571. Several approaches to sharpen these bounds are discussed. In particular, a variational problem is considered, in which p is supposed to have a certain compactly supported convolution root, and which leads to an improved upper estimate, Λ < 0.8567... The paper closes with a proposal for a multivariate analogue of the uncertainty relation.