Distributional Calculus Part 4: Properties of Distributions

So, in the previous post, we found that distributions gave an alternate way to characterize functions; that is, by mapping from a set of test functions instead of from a compact set $X$. Test functions turn out to be completely central to how operations are performed! In fact, I'll spoil the entire content of this post by saying any operation on $f$ can be 'moved' to apply on the set of test functions instead.

But before that, let's list some basic properties which are more evocative of elementary real analysis than anything else. For a distribution $\langle f, \phi\rangle$:

• Linearity, i.e. $f(a\phi_1+\phi_2)$ for any real constant $a$ and test functions $\phi_1,\ \phi_2$;
• There exists a sequence of test functions $\{\phi_n\}$ such that $\phi_n \to f$

All of these properties are necessary, but we'll be making the most use out of the second one. Recall the super-useful integral characterization of a distribution $$T_f(\phi)=\int_{\mathbb{R}}f(x)\phi(x)\,dx?$$ That can only be expressed if $f$ is a function with no weird generalized properties. Yet now, if we consider $f$ as the limit of a sequence of test functions, $\phi_n$ is a classically defined function for all $n$ and it is now possible to write  $$T_f(\phi)=\lim_{n\to\infty}\int_{\mathbb{R}}\phi_n(x)\phi(x)\;dx$$ for any generalized function $f$.* Great! Now we can look at any and all distributions the easy way.

The real magic starts when we attempt to translate the distribution. Recall that any function can be translated $y$ units by taking $f(x-y)$ instead of $f(x)$; the same thing can be done for generalized functions by considering $\lim_{n\to\infty}\langle \phi_n(x-y),\phi(x)\rangle$. (Let's define the translation function tau as $\tau_y\phi(x)=\phi(x-y)$.) Using some simple $u$-substitution magic, \begin{align}\langle\tau_yT_f,\phi\rangle &=\lim_{n\to\infty}\langle \phi_n(x-y),\phi(x)\rangle\\&=\lim_{n\to\infty}\int_{\mathbb{R}}\phi_n(x-y)\phi(x)\;dx;\qquad u = x-y\\&=\lim_{n\to\infty}\int_{\mathbb{R}}\phi_n(u)\phi(u+y)\;du\\&=\langle T_f,\tau_{-y}\phi\rangle.\end{align} We have essentially found that any distribution can be translated by applying the opposite translation to every test function in $\mathcal{D}$. To reiterate:$$\langle\tau_yT_f,\phi\rangle =\langle T_f,\tau_{-y}\phi\rangle.$$ Hooray!

Differentiating a distribution works in much the same way as translation in that the operation gets pawned off onto the test function but with an extra minus sign. However, it does involve an extra technique: integration by parts. I assume that nobody who is reading this is unfamiliar with the practice, but, for the sake of cute mnemonics, a friend of my fiancé's refers to $$\int u\;dv = uv - \int v \;du$$as "sudv uv svidoo."

Let's take a moment to appreciate how adorable that is.

The actual fancy differentiation trick can be proved in essentially one integration-by-parts step:\begin{align}\left\langle \frac{d}{dx}T_f,\phi\right\rangle &= \lim_{n\to\infty}\int_\mathbb{R}\left(\frac{d}{dx}\phi_n(x)\right)\phi(x)\;dx\\&= \lim_{n\to\infty}-\int_\mathbb{R}\phi_n(x)\left(\frac{d}{dx}\phi(x)\right)\;dx \\ &=\left\langle T_f, -\frac{d}{dx}\phi(x)\right\rangle\end{align}(brownie points if you've already figured out what happened to the $uv$ term). This identity is essential for a crazy number of distributional calculus proofs.

For example, we can directly use this identity to prove the Dirac delta function is the distributional derivative of the Heaviside function in two seconds. Let $T_H= \int_0^\infty \phi(x)\,dx$ represent the Heaviside distribution. Now, from the above identity, we conclude $$\left\langle \frac{d}{dx}T_H(x),\phi\right\rangle=\left\langle T_H(x),\frac{d}{dx}\phi\right\rangle=-\int_0^\infty \phi(x)\,dx=\phi(0)-\phi(\infty)=\phi(0),$$ that is, because $\phi$ is zero at infinity. Yet $\phi(0)=\langle \delta, \phi\rangle$ by definition! We're done here.

As super awesome as that is, there should be some material on how all this pertains to weak solutions of DEs up on Thursday. Woooo! This is basically my definition of a party!



* The MCT happened here. Shhh.

Distributional Calculus Part 3: Distributions

Sorry for the delay, guys! I just started a rather demanding full-time job, so it may be a bit hard to keep up the quality of these posts. Let's hope it gets easier...

Today brings us to the most important definition in distributional calculus: the distributions themselves.

Here's the formal definition using the set of test functions $\mathcal{D}(\mathbb{R})$ we defined earlier:

Any linear functional $T: \mathcal{D}(\mathbb{R}) \to \mathbb{R}$ a distribution. In addition, for a locally integrable function $f(x):X\to\mathbb{R}$, a corresponding distribution can be defined by $$T_f(\phi)=\int_{\mathbb{R}}f(x)\phi(x)\;dx.$$We usually write $\langle T, \phi\rangle$ instead of $T(\phi)$ and call the set of all distributions of this type $\mathcal{D}'(\mathbb{R})$.

There are only two things needed to truly understand this definition; how to take the average of a continuous function and what test functions are. Check out the integrand. Multiplying the target function $f(x)$ by each individual test function $\phi(x)$ has the effect of scaling $f(x)$ at every point---in particular, the integrand zeros out outside the support of $\phi(x)$, while the other points are weighted depending on $\phi(x)$. Hence every individual component of the definition a weighted average of $f(x)$ over a compact set. (Strichartz directly compares this to finding the temperature of a room with a thermometer: it won't display the temperature at one point, rather the average temperature of some portion of the area.) If each of these weighted averages are known for every existing $\phi(x)$, that is what defines the distribution.

Defining distributions in this way lets us account for objects that we think look like functions, but actually aren't. The Dirac delta function is the perfect example---the infinite value at zero ruins anything, so it isn't really a function*. However, the integral of $\delta(x)$ is bounded no matter what test function we weight it by, so the 'average' exists over every possible range, meaning $\delta(x)$ is a distribution. In particular, $$\langle \delta,\phi\rangle = \phi(0).$$

It would be useful to go over a couple useful properties of distributions, starting with the issue of consistency. This was supposed to happen today! Unfortunately, I'm dead tired and need to go lie down forever. Let's leave the important properties for next week.



* The Dirac delta function is to functions what killer whales are to whales... a complete misnomer.

Distributional Calculus Part 2: Compact support and test functions

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.The following is written assuming an audience who cares or wants to care about mathematical formality but needs some intuitive background in order to learn quickly.


A few minor definitions are needed to understand what distributions represent. We define a set $X$ and function $\phi: X\to\mathbb{R}$ for the rest of this post.

The first two definitions are very simple.


Definition: Suppose $\phi$ is in $L_p(\mathbb{R}^n)$ and $X$ is open. We say $\phi$ is locally integrable if, for all compact subsets $A$ of $X$,
$$\int_A |\phi(x)|\;dx< \infty.$$The space of all such functions is called $L_p^{loc}$.


The formal definition of compactness can be found here. For those who haven't studied real analysis, a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded.

Definition: The support of $\phi$, written supp($\phi$), is the closure of the set of points in X where f is non-zero. That is,

$$\operatorname{supp}(\phi) = \{x\in X \,|\, \phi(x)\ne 0\}.$$(Topologists use a slightly different definition.)

From here, a slightly more specific property can be considered:

Definition: A function $\phi$ is said to have compact support if $supp(\phi)$ is compact.

It's hard to come up with a compactly supported function without specifying that the complement of the support is zero. As a result, most easily representable test functions, even the continuous and infinitely differentiable ones, are defined piecewise. We consider a few examples.

Note that compact support can also be interpreted as the function vanishing outside a compact set; continuous functions are always nonzero on an open set, so taking the closure in the definition of support is necessary.

One of the simplest examples of a compactly supported function is $\chi_A(x)$, where $A$ is a compact set and
$$\chi_A(x)=\left\{\begin{array}{ll}1&x\in A\\0& x \notin A.\end{array}\right.$$This is the identity on $A$ and zeros out everything else. In fact, the composition of $\chi_A(x)$ with any function on $x$ will have compact support as well. Here are a couple examples:

(Test yourself! Is H(x) from the previous post compactly supported? Are B-splines?)

This example leads well into the last definition.

Definition: A function $\phi$ is a test function if  it has compact support and is infinitely differentiable (i.e., in $C^\infty$). We refer to the space of all test functions on a set $X$ as $\mathcal{D}(X)$.


This is a crucial definition! It's weird for a function to have compact support but to also be infinitely differentiable, so let's generate a couple examples. Consider
$$\psi(x)=\left\{\begin{array}{ll}e^{-\frac{1}{1-x^2}}&|x|<1\\0& |x|\geq 1.\end{array}\right.$$This is a lot smoother than the previous function, and looks like a bump:

A slightly more complicated example would be
$$u_A(x)=\int_{\mathbb{R}^n}\chi_A(x-y)u(y)\;dy$$where $u(x)$ is a locally integrable function in $\mathbb{R}^n$ (the technique used to generate this example is called Sobolev's mollification method). If you're familiar with convolution already, it should not be difficult to prove this function is compactly supported. It looks like someone built a sandcastle shaped like a regular $\chi$ function and a wave rolled over it:

Many operations, such as translation and scaling, preserve infinite differentiability and compact support. Linear combinations of test functions and products of test functions are also test functions themselves.

(Test yourself! Can test functions be analytic?)


So, our point---these definitions are necessary in order to understand what distributions are. We'll go into this in detail next week.

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.