Functional Analysis (11): Distributions and Sobolev Spaces — Generalized Solutions

Thu, 21 Oct 2021 09:00:00 +0000

I want to start with a confession. For years I treated the Dirac delta the way an undergraduate physicist does: as a function that is zero everywhere except at the origin, where it is infinite, and whose integral is one. That description is, of course, mathematical nonsense. No measurable function can have those properties. Yet every quantum mechanics textbook uses $\delta$ on page one, every signal processing course writes $\delta(t)$ for an impulse, and every PDE book invokes Green’s functions $$E$$ satisfying $\Delta E = \delta$ . Either an entire scientific community has been making a fundamental error for a century, or there is a way to make this object rigorous. The latter, obviously — and the way is the theory of distributions.

Sobolev-Spaces on Chen Kai Blog

Functional Analysis (11): Distributions and Sobolev Spaces — Generalized Solutions