Lecture 10, Fourier transform (2) and its applications

On December 12, we covered the following:

1. an equivalent definition of $W^{k,2}$, via Fourier transform (as in Evans in Chapter 5.8),
2. an applition of Fourier transform to the heat equation in a half-space, including a derivation of an explicit formula for solutions, and a discussion of the properties of solutions (smoothness, continuity at $t=0$ at all points where the initial data is continuous, infinite speed of propagation etc.); this material can be found in Evans, Sections 4.3.1(b) and 2.3.1.