User:Yannis
From TorontoMathWiki
Some thoughts on topics presented during the lectures
- Lectures 1 & 2
In those first lectures the important Uncertainty Principle was described. The mathematical proof of it is quite simple and can be found in Tao's notes for example. A related Uncertainty Principle is the so called Hardy Uncertainty Principle. Precisely it states that if $f$ satisfies $|f(x)| \leq c_1 e^{-\pi x^2}$ and $|\hat{f} (\xi)| \leq c_2 e^{-\pi {\xi}^2}$ for all $x, \xi$ and some constants $c_1, c_2$ then $f$ is a constant multiple of the Gaussian $e^{-\pi x^2}$. The proof of this theorem of Hardy was presented in his paper "A Theorem Concerning Fourier Transforms" and it uses complex analysis, all that someone needs is a modification of the Phragmen-Lindelof principle so to apply it with squares on the exponents. Note that the theorem implies (actually someone has to trace this through its proof) that if $|f(x)| \leq c_1 e^{- a \pi x^2}$ and $|\hat{f} (\xi)| \leq c_2 e^{- b \pi {\xi}^2}$ then if $ab > 1$ we have that $f \equiv 0$. Recently interest has been shown in finding a real variable proof of this theorem of Hardy and this was done by Cowling, Escauriaza, Kenig, Ponce and Vega [1]. I would like to learn more about this, for example what is the motivation behind this new proof? There seems to be a connection with the Schrodinger equation as the same team minus Cowling has published a series of papers on a Schrodinger version of the Hardy Uncertainty Principle.
Moreover there was some discussion on the inequality of Bernstein. The proof towards which Pr. Colliander gave us hints involves interpolation and rescaling. If someone wants to see a slightly different version of this proof then take a look at pages 29-30 of the notes of T. Wolff on Harmonic Analysis [2] (in pages 31-33 a couple of related results to Bernstein's inequality are presented). Wolff makes the clever observation that a function (in $L^2$ for instance) supported in a disk can be expressed alternatively as a convolution of a dilated version of a Schwartz function and itself. Then the result follows by Young's inequality (which involves interpolation in its proof, so in some sense the proof again follows by rescaling and interpolation).
