Paweł Strzelecki | Lecture 9, Isoperimetric problem in the plane and Fourier transform (1)

On December 5, we first discussed a solution of the classic isoperimetric problem in the plane, using Stokes’ theorem, Fourier series and the Parseval identity.

Then, we have defined the Fourier transform as

$\hat f (\xi) = \int_{\mathbb{R}^n} f(x) \exp (2\pi i x\xi)\, dx$

for $f$ in the class of Schwartz rapidly decreasing functions (note: the definition makes sense also for integrable $f$ ), proved the elementary algebraic properties of $f\mapsto \hat f$ and the Fourier inversion theorem on the class of Schwartz functions. Remark: to make the proof precise, one needs to know what is the Fourier transform of at least one function; for Schwartz functions, a natural choice is $f(x)=\exp (-\pi \vert x \vert^2)$ which turns our to be a fixed point of $f\mapsto \hat f$ .