# Ajay Kumar (University of Delhi)

When:
July 14, 2020 @ 6:00 am – 8:00 am
2020-07-14T06:00:00-10:00
2020-07-14T08:00:00-10:00
Where:
Lecture held in Elysium

Title: Uncertainty Principles on Locally Compact Groups
by Ajay Kumar (University of Delhi) as part of Topological Groups

Lecture held in Elysium.

Abstract
Some of the uncertainty principles on $mathbb{R}^n$ are as follows:

Qualitative Uncertainty Principle: Let $f$ be a non-zero function in $L^1(mathbb{R}^n)$. Then the Lebesgue measures of the sets ${x: f(x) eq 0 }$ and ${xi : widehat{f}(xi) eq 0}$ cannot both be finite.

Hardy’s Theorem: Let $a,b,c$ be three real positive numbers and let $f: mathbb{R}^n to mathbb{C}$ be a measurable function such that

(i) $|f(x)| leq cexp{(-api |x|^2)}$, for all $x in mathbb{R}^n$
(ii) $|widehat{f}(xi)| leq cexp{(-bpi |xi|^2)}$, for all $xi in mathbb{R}^n$.

Then following holds:
If $ab>1$, then $f=0$ a.e.
If $ab =1$, then $f(x)= alpha exp{(-api |x|^2)}$ for some constant $alpha$.
If $ab< 1$, then there are infinitely many linear independent functions satisfying above conditions.

Heisenberg Inequality: If $f in L^2(mathbb{R}^n)$ and $a,b in mathbb{R}^n$, then

$$left( int_{mathbb{R}^n}|x-a|^2|f(x)|^2 dx right) left( int_{mathbb{R}^n}|xi-b|^2|widehat{f}(xi)|^2 dxi right) geq frac{n^2|f|^4}{16pi^2}.$$
Beurling's Theorem: Let $f in L^1(mathbb{R}^n)$ and for some $k(1leq kleq n)$ satisfies
$$int_{mathbb{R}^{2n}} |f(x_1, x_2, dots , x_n)||widehat{f}(xi_1, xi_2, dots , xi_n)|e^{2pi |x_kxi_k|} dx_1dots dx_n dxi_1dots dxi_n< infty.$$
Then $f = 0$ a.e.

We investigate these principles on locally compact groups, in particular Type I
groups and nilpotent Lie groups for Fourier transform and Gabor transform.