Unlocking Distribution Theory: Understanding Generalized Functions & derivatives

From Smooth Functions to Distributions: What Happens When You Differentiate a Functional?

Introduction: More Than Just Derivatives

If you've ever taken a calculus class, you know how to differentiate a function. But what if you're not differentiating a function—but a functional? Even more mind-bending: what if the object you're working with isn't even a function in the traditional sense, but a generalized function or distribution?
Welcome to the magical world of distribution theory, where even the Dirac delta "function" makes perfect sense, and derivatives can be defined in a way that bypasses all the usual problems of discontinuities and infinite spikes.
In this post, we'll explore how the derivative of a distribution is defined, why this works, and how it connects to the test function space S, a smooth and rapidly decaying space that serves as the playground for this theory.
We'll also give you a sneak peek into how you can explore these ideas computationally using SageMath.

The Space S: Our Testbed of Functions

Let’s begin with the space 𝑆, the Schwartz space. It consists of all functions 𝜙(𝑥) that are:

• Infinitely differentiable : \( \varphi^{(k)}(x) \) exists for all 𝑘 ,
• And decay faster than any power of \( \frac{1}{x} \) : for all k,q, \[ \sup_{x \in \mathbb{R}}|x^q \varphi^{(k)}(x) | < \infty, \]

These are the smoothest, fastest-decaying functions you can imagine. They're ideal for testing how nice (or nasty) other objects behave under limits, differentiation, and integration. Hence, the name test functions.

Dual Space 𝑆′: Enter the Distributions

Now consider 𝑆′, the space of tempered distributions. This is the space of continuous linear functionals on 𝑆 .If \( f \in S' \),it means you can pair it with any test function \(\varphi \in S \) via: \[ \langle f, \varphi \rangle \] which is a number that depends linearly and continuously on \(\varphi\).
And here's the wild part: many "functions" that aren’t actual functions live here. The Dirac delta 𝛿(𝑥), for instance, is not a function, but it acts on test functions via: \[ \langle \delta, \varphi \rangle=\varphi(0) \]

Derivatives of Distributions

Here’s the big question: If \( f \in S'\) , can we define its derivative \( f'\) in a way that still makes sense?

Yes! And we do it by defining the derivative by how it acts on test functions. Specifically, we define: \[ \langle f', \varphi \rangle := -\langle f, \varphi' \rangle \] for all \( \varphi \in S\)

This definition mirrors integration by parts, but without worrying about boundary terms (since test functions decay to zero at infinity). And the magic? This definition keeps \(𝑓^′\) inside \(𝑆^′\)—it’s still a tempered distribution!
This also aligns perfectly with how we differentiate weak solutions to PDEs.

Slowly Increasing Functions and Extensions

There's another class of functions, called slowly increasing functions—those that grow no faster than a polynomial. These functions live in a space \(K \subset S' \).

The amazing thing is : Every slowly increasing function defines a tempered distribution.
And so do all of its derivatives!
This means you can go back and forth between functions and distributions smoothly—pun intended!

💡 Try It Yourself! Now You can copy and paste directly into here Run SageMath Code Here