Generalized Functions: Definition, Theory & Applications in Mathematics, Physics & Engineering

Generalized Functions Explained — What Are They?

Have you ever tried describing a moment so brief, it's like it only exists at a single point in time—like a camera flash? That’s what generalized functions (aka distributions) do in math.

They extend the idea of ordinary functions to include strange but useful objects—like the delta function, which isn’t a real function at all in the usual sense.

Why Use Generalized Functions?

Classical functions struggle with sharp spikes or sudden impulses. For example, how do you model:

• A hammer strike (force at a single moment)?
• A spark (a single flash in time)?
• A point charge in physics?

👉 Generalized functions let us define and manipulate such phenomena rigorously using calculus.

Core Concept

A generalized function is a rule that takes in a test function φ(x) (a smooth, well-behaved function) and returns a real number.

We don’t focus on values at individual points. Instead, we define everything in terms of how the generalized function acts on φ(x):

\[ (f, \varphi) = \text{some real number} \]

This “pairing” must follow two basic rules:

Key Properties

1. Linearity
If you scale and add test functions, the response is linear: \[ (f, \alpha_1 \varphi_1 + \alpha_2 \varphi_2) = \alpha_1 (f, \varphi_1) + \alpha_2 (f, \varphi_2) \]

2. If your test functions approach zero, so should the result: \[ \varphi_n \to 0 \Rightarrow (f, \varphi_n) \to 0 \]

🎯 Examples in Action

✅ Regular Generalized Function

If f(x) is a normal, integrable function, we define:

\[ (f, \varphi) = \int f(x) \varphi(x) , dx \]

This is regular because it comes from an actual function.

The Delta Function δ(x)

This famous example isn’t a true function—it’s purely a generalized function.

\[ (\delta, \varphi) = \varphi(0) \]

Think of it like a perfect sensor that picks out the value at x = 0. It has no width or shape—it’s like a mathematical needle or a snapshot in time.

Shifted version: \[ (\delta(x - x_0), \varphi(x)) = \varphi(x_0) \]

Regular vs. Singular Distributions

• Regular: Comes from actual functions (e.g., f(x) = sin(x), 1, e^x)
• Singular: Does not come from real functions — e.g., δ(x), derivatives of δ(x)

Even constants can be generalized functions:

\[ (1, \varphi) = \int \varphi(x) , dx \]

Visualization Tip

Imagine a series of smooth test functions φₙ(x) that get narrower and taller, centered at 0. No matter how small, the delta function always "sees" what’s happening at that exact point.

• In math software like SageMath or Python (with SymPy), you can simulate this effect to better visualize δ(x).

The Bigger Picture

Generalized functions live in a mathematical space called K′ (the dual of the space of test functions K). Regular functions are just a special case.

When you see expressions like:

\[ \delta(x) \varphi(x) , dx \]

…it’s shorthand for the more abstract idea: \( (\delta, \varphi) = \varphi(0) \)

Delta Function Approximation in SageMath

We'll use a family of Gaussian functions:

\[ \varphi_n(x) = \frac{1}{\pi n} \cdot e^{-\left(\frac{x}{n}\right)^2} \]

These get narrower as 𝑛→0, but always integrate to 1 — a good model for δ(x).

What This Shows:As 𝑛 gets smaller:

• The function gets sharper and taller
• It concentrates more around 𝑥=0
• But the area under the curve stays ≈ 1, simulating δ(x)

Want More?You could also explore:

• Using other approximations like rectangular pulses or sinc functions
• Plotting how each approximation acts on a test function( e.g., \( \varphi(x) = \sin(x) \)