Alladi Sitaram

The Uncertainty Principle: A Mathematical Survey

We survey various mathematical aspects of the uncertainty principle, including Heisenberg’s inequality and its variants, local uncertainty inequalities, logarithmic uncertainty inequalities, results relating to Wigner distributions, qualitative uncertainty principles, theorems on approximate concentration, and decompositions of phase space.

Uncertainty principles on certain Lie groups

AbstractThere are several ways of formulating the uncertainty principle for the Fourier transform on ℝn. Roughly speaking, the uncertainty principle says that if a functionf is ‘concentrated’ then its Fourier transform

Local uncertainty inequalities for locally compact groups

\tilde f

A QUALITATIVE UNCERTAINTY PRINCIPLE FOR SEMISIMPLE LIE GROUPS

cannot be ‘concentrated’ unlessf is identically zero. Of course, in the above, we should be precise about what we mean by ‘concentration’. There are several ways of measuring ‘concentration’ and depending on the definition we get a host of uncertainty principles. As several authors have shown, some of these uncertainty principles seem to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including ℝn, the Heisenberg group, the reduced Heisenberg groups and the Euclidean motion group of the plane.

The injectivity of the pompeiu transform andL p -analogues of the Wiener-Tauberian theorem

We formulate analogues of the Hausdor-Young and Hardy- Littlewood-Paley inequalities, the Wiener Tauberian theorem, and some un- certainty theorems on Riemannian symmetric spaces of noncompact type using the Helgason-Fourier transform.

Injectivity sets for spherical means on ℝ n and on symmetric spacesand on symmetric spaces

Let G be a locally compact unimodular group equipped with Haar measure m, G ^ its unitary dual and μ the Plancherel measure (or something closely akin to it) on G ^ . When G is a euclidean motion group, a noncompact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given θ∈[O,½) there exists a constant K θ such that for all ƒ in a certain class of functions on G and all measurable E ⊆ G ^ , (∫ E Tr(π(ƒ) ∗ π(ƒ))dμ(π) ½ ≤ K θ μ(E) θ ||Φ θ ƒ|| 2 where Φ θ is a certain weight function on G (for which an explicit formula is given). When G=R k the inequality has been established with Φ θ (x)=|x| kθ .

Local uncertainty inequalities for compact groups

Abstract It is well known that if the supports of a function f ϵ L1(Rd) and its Fourier transform \ tf are contained in bounded rectangles, then f = 0 almost everywhere. In 1974 Benedicks relaxed the requirements for this conclusion by showing that the supports of f and \ tf need only have finite measure. In this paper we extend the validity of this property to a wide variety of locally compact groups. These include Rd × K, where K is a compact connected Lie group, the motion group, the affine group, the Heisenberg group, SL(2, R), and all noncompact semisimple groups with some additional restrictions on the functions f.

The Helgason Fourier Transform for semisimple Lie groups I: the case of SL 2 (R)

Recently M. Benedicks showed that if a function f ∈ L 2 (R d ) and its Fourier transform both have supports of finite measure, then f = 0 almot everywhere. In this paper we give a version of this result for all noncompact semisimple connected Lie groups with finite centres.