Local Properties of Generalized Functions: Essential Points, Vanishing, and Support in Distribution Theory

Local Properties of Generalized Functions

Introduction

Generalized functions—also known as distributions—extend the concept of classical functions to handle cases where traditional tools break down. They were developed to rigorously define operations like differentiation on non-smooth functions and to solve differential equations involving singularities. Unlike ordinary functions, generalized functions cannot be evaluated at isolated points, which brings us to a crucial concept in their theory: local properties.

Why Generalized Functions Differ from Ordinary Functions

Ordinary functions assign values to individual points. For example, we might write 𝑓(2)=3. But with generalized functions, such a pointwise definition doesn't make sense. Instead, they are defined through their action on a class of smooth test functions with compact support—denoted by \( \varphi(x) \in K \).

As a result, statements like “a generalized function is zero at \( {x_0}^" \) are meaningless. But saying “a generalized function is zero in a neighborhood of \( {x_0}^" \) can be made precise and useful.

Vanishing and Essential Points

Vanishing in a Neighborhood

We say that a generalized function 𝑓 vanishes in a neighborhood 𝑈 of \( x_0 \) if

\[ \langle f, \varphi \rangle = 0 \quad \text{for all } \varphi \in K \text{ with support in } U. \]

This aligns with the intuition from ordinary functions. For example, if a function 𝑓(𝑥) is zero almost everywhere in 𝑈, then the generalized function it defines also vanishes in 𝑈.

The delta function \( \delta(x - x_0) \), which is singular at \( x_0 \), vanishes in every neighborhood that does not contain \( x_0 \) .

Essential Points

If a generalized function 𝑓 does not vanish in any neighborhood of a point \( x_0 \),then \( x_0 \) is called an essential point of 𝑓.

Take \( f(x)=x^2 \) . The function vanishes at 𝑥=0, but since it does not vanish in any neighborhood of 0, the point 𝑥=0 is still an essential point. In fact, every point in \( \mathbb{R} \) is essential for this function because it does not vanish on any open interval.

Support of Generalized Functions

The support of a generalized function 𝑓 is the closure of the set of all its essential points. This is analogous to the support of an ordinary function, but generalized to accommodate singularities.

• For a regular generalized function (one associated with a continuous or piecewise continuous function), the support is just the usual support of that function.
• The support of the delta function \( \delta(x - x_0) \) is the single point \( x_0 \)

If 𝑓 vanishes in a neighborhood of every point in an open set 𝐺 , we say that 𝑓 vanishes on 𝐺. And if 𝑓 vanishes in a neighborhood of every point in \( \mathbb{R} \), then 𝑓=0 in the distributional sense.

Local Comparison of Generalized Functions

Two generalized functions 𝑓 and 𝑔 are said to coincide on an open set 𝐺 if 𝑓−𝑔 vanishes on 𝐺.

A fundamental result here is:

• If two generalized functions coincide in a neighborhood of every point, then they are identical as generalized functions.

This theorem underlines a powerful fact: generalized functions are determined entirely by their local behavior.

Constructing Generalized Functions from Local Properties

One might ask: what if we start with an ordinary function 𝑓(𝑥) that is not even locally summable (like 1/𝑥)? Can we still define a generalized function that “agrees” with 𝑓(𝑥) wherever possible?

The answer is yes—under certain conditions. There exist ways to define generalized functions from such singular ordinary functions so that key operations like addition, multiplication by smooth functions, and differentiation are preserved.

This is crucial for extending classical analysis to handle real-world phenomena where singularities occur—like point charges in physics or impulses in signal processing.

Applications and Importance

Generalized functions are not just theoretical curiosities. They are indispensable in:

• Signal processing: The delta function models instantaneous signals or impulses.
• Physics: Distributions model point sources and discontinuities in fields.
• Partial Differential Equations (PDEs): Weak solutions to PDEs often require distributions to make sense.

Further Reading

To dive deeper into this subject, check out:

• L. Schwartz, Théorie des distributions
• V. S. Vladimirov, Generalized Functions in Mathematical Physics
• I. M. Gelfand & G. E. Shilov, Generalized Functions Vol. 1

Let's check this out in SageMath now!

Vanishing and Essential Points

You can numerically verify this using the delta function and a smooth test function:

Interpretation: The integral is 0, confirming that the delta function has no effect outside its singularity at 𝑥=1, and thus vanishes in neighborhoods that exclude that point.

Essential Points

If 𝑓 does not vanish in any neighborhood of \( x_0 \), then \( x_0 \) is called an essential point of 𝑓.
Consider \( 𝑓(𝑥)=𝑥^2 \). Although 𝑓(0)=0, does it vanish around 0?

Analysis: \( 𝑓(𝑥)=𝑥^2 \) vanishes at 𝑥=0 but is nonzero in any surrounding neighborhood, confirming that 𝑥=0 is still an essential point.

Local Equivalence of Generalized Functions

Let's compare \[ f(x) = x^2 \] \[ g(x) = x^2 + e^{-x^2} \]

Explanation: If the two integrals differ, then 𝑓 and 𝑔 do not coincide in the sense of generalized functions.