The Uncertainty Principle
The famous Heisenberg Uncertainty Principle from Quantum Mechanics has a striking formulation in the language of Harmonic Analysis. It essentially says that a non-zero measure (distribution) and its Fourier transform cannot be simultaneously small. Throughout the years this broad statement raised a multitude of deep mathematical questions, each corresponding to a particular sense of ``smallness.'' The majority of these questions were inspired and are closely connected to many fundamental problems in approximation theory, inverse spectral problems for differential operators and Krein's canonical systems, classical problems in the theory of stationary Gaussian Processes, signal processing, etc. Many of these problems remained open for more than half a century, and some of them were even considered completely intractable.
During Fall 2012 - Spring 2013 the lectures for the Internet Analysis Seminar will be prepared by Professor Alexei Poltoratski.