Monday, March 2, 2015

Distributional Calculus Pt. 1: What is it?

In high school, despite being told I was "good at math" for being able to perform simple algebra, I was terrified of calculus. It was a scary word---"calculus"---and I didn't want to be outed as an impostor who wasn't ever good at math at all. That's how I ended up enrolled in the easiest calculus course offered at my high school, a place where most people took AP Calc. That's also how I ended up bored with the slow pace and lack of formality of my first calculus course, and transferred to AP Calc halfway through the year. That's also when I developed the unmitigated desire to become a mathematician; the calculus floodgates had been opened, and the only cure was more calculus. Calculus was followed by real analysis. Real analysis was followed by functional analysis.

Which brings us here... to the ultimate form of calculus. But why? Why does such a thing exist?

The catalyst for developing a more general form of calculus came when some people, such as physicists and engineers, decided it was okay to consider derivatives of non-differentiable functions. We consider the Heaviside step function ($H(x)$) as the quintessential example: this function is constant and hence has a zero derivative everywhere except at the jump discontinuity, where the classical definition of the derivative breaks down. One could reason that, because the derivative at a point is the slope of the tangent line, and the tangent line at the jump is a vertical line with infinite slope, $H'(0)$ is infinity. We therefore understand the derivative of the Heaviside function to be zero everywhere except at the jump, where it's infinite. That's the Dirac delta function ($\delta(x)$)!

Generally---and I apologize for stereotyping here---generally, physicists and engineers are totally okay with this interpretation and accept it as fact, but mathematicians are upset by the hand-waving. It particularly bothered Sergei Sobolev and Laurent Schwartz, whose work lead to the first mathematical justification of these ideas. This formalization of the engineers' and physicists' approaches grew to be called distributional calculus.

Distributions (also called generalized functions) define a broad set of function-like objects including, but not limited to, classical functions (hence, generalized functions). Distributional calculus is the study of calculus on this larger class of objects. This certainly allows for a formal reimagining of the Heaviside example given above: the Heaviside function is nondifferentiable at a point, but its distribution is differentiable everywhere! It can also be used to describe "weak" solutions of DEs. So, if you're like me and can't get enough calculus, it's just... more. More calculus.

Distributional calculus is also a great demonstration of the central public-relations conflict of real/functional/complex analysis: it's both the coolest thing anyone has done, ever, but also completely inaccessible to laypeople. In particular, the notation gets very intimidating, very fast. (Converting any idea from functions to distributions requires several million extra symbols.)

Our goal with this series is to provide a resource for basic distribution theory that includes all of the formal definitions, justifications and theorems with as little hand-waving as possible, while also fully explaining these definitions through appeals to intuition. There are already great books that deal with the formal side of distribution theory (Haroske and Triebel, 2008; Friedlander and Joshi, 1998) and great books that eschew formality in order to be accessible to physicists and engineers (Strichartz, 2003). These books are much better than a series of blog posts---that's why the authors of the books get paid. However, we adopt a different approach for our audience: the first set of textbooks caters to analysts, the second to people who don't care for analysis, while we assume the audience cares or wants to care about mathematical formality but needs some intuitive background in order to learn quickly.

Without further exposition, here's the game plan for March:

  • Week 1 & 2: Basic definitions (compact support, test functions, distributions, distributional derivatives, all that good stuff)
  • Week 3: The big examples
  • Week 4: A couple important theorems
  • Week 5 (March 31st): Recent papers /books for suggested further reading

Lastly, especially if you're a non-mathematician who doesn't care about overt formality, I cannot recommend the Strichartz enough. It's hilarious! I definitely got something out of it despite being peeved at the lack of formal analysis.

[1] Haroske, Dorothee, and Hans Triebel. Distributions, Sobolev spaces, elliptic equations. European Mathematical Society, 2008.
[2] Friedlander, Friedrich Gerard, and Mark Suresh Joshi. Introduction to the Theory of Distributions. Cambridge University Press, 1998.
[3] Strichartz, Robert S. A guide to distribution theory and Fourier transforms. Singapore: World Scientific, 2003.

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