Questions March 24

Questions to think about while reading the course material:

What is the difference between a weak derivative and a distributional derivative?
How does one define the product of a function and a distribution? What conditions are needed on the function?
On p. 43 of Wahlén it is said that $LaTeX: g\left(x\right)\delta_0\left(t\right)$ $g\left(x\right)\delta_0\left(t\right)$ and $LaTeX: \delta_0\left(x\right)\delta_0\left(t\right)$ $\delta_0\left(x\right)\delta_0\left(t\right)$ are actually not well-defined. Why not?
Why do we need another class of test functions and distributions in order to define the Fourier transform?
Can you think of a second order linear PDE with constant coefficients which is neither elliptic, hyperbolic, nor parabolic?
In Example 2 on pp. 257-258 it is shown that the function u has no weak derivative. Does it have a distributional derivative and, if so, what is it?