Delta Dirac: Showing it's a Distribution

In summary, the Dirac delta function is a distribution defined as a functional that maps a space of functions to the real line by evaluating the function at 0. This definition makes it a continuous and linear function, with linearity demonstrated by the linearity of integration and continuity demonstrated by its boundedness. This allows for the generalization of functions and the representation of vectors in terms of their actions on other vectors.
  • #1
sdickey9480
10
0
I realize it's not a function in the classical sense, but how would one show that the delta dirac function is a distribution i.e. how do I show it's continuous and linear given that it's not truly a function?
 
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  • #2
What is your definition of the Dirac delta function??
 
  • #3
Given any x_0 ∈ Rn, the delta function is the distribution, δ_{x_0} :D(Rn)→C
given by the evaluation of a test function at x_0: ⟨δ_{x_0} , φ⟩ = φ(x_0)
 
  • #4
The set of set of test functions is a vector space. The reals are are also a vector space. Use that for linearity.
Test functions are smooth, use that for continuity.
 
  • #5
Hi sdickey9480!

I think your question really amounts to 'what is a distribution'? As mentioned by the previous posters it has to do with test functions, and more generally with special vector spaces (usually complete ones and those endowed with a norm).

An amazing triumph of functional analysis is representing vectors (in this case non pathological functions) in terms of their actions on other vectors. By action on other vectors, I mean given any vector [itex]v[/itex] in the vector space [itex]V[/itex], define a mapping [itex]\hat{v}:V\rightarrow \mathbb{R}[/itex]. This mapping is given by the Riesz Representation Theorem, and in our case it means [itex]\hat{v}(g):=\int fg \mathrm{d}x[/itex]

[itex]\hat{v}[/itex] is called linear because [itex]\hat{v}(f+g)=\hat{v}(f)+\hat{v}(g)[/itex].

The second important property we want [itex]\hat{v}[/itex] to have is that of continuity. Another surprising result of functional analysis says that a functional (any linear map from the vector space into the reals, like [itex]\hat{v}[/itex] for example) is continuous if and only if it is bounded in the operator sense. That is, [itex]\hat{v}[/itex] is bounded if and only if sup[itex]\{\hat{v}(f): ||f||_\infty = 1 \}< \infty [/itex]

What I have done is built the necessary machinery to generalize functions. What I have shown is that any vector (or in this case non-pathological function) can be thought of as a continuous linear functional. A distribution is then just one of these continuous linear functionals.

So to answer your question, the dirac delta function [itex]\delta[/itex] is defined as a functional, mapping some space of functions to the real line by [itex]\delta (f) = \int f\delta \mathrm{d}x := f(0)[/itex]. It is clear that [itex]\delta[/itex] is linear because integral is linear (actually strictly speaking the integral doesn't make sense, hence the need for generalized functions to begin with. We really define [itex]\delta[/itex] to be linear).

Why is [itex]\delta[/itex] continuous? Because if [itex]||f||_\infty = 1[/itex] and [itex]f[/itex] is continuous, then [itex]f(x)\leq 1[/itex] for any [itex]x[/itex]. Hence [itex]\delta[/itex] is bounded by [itex]1[/itex], and therefore continuous.
 

What is the Delta Dirac distribution and what does it represent?

The Delta Dirac distribution, also known as the Dirac delta function, is a mathematical concept used to represent an infinitely tall and narrow spike at a specific point on a real number line. It represents a generalized function that is zero everywhere except for its peak at the specified point.

How is the Delta Dirac distribution defined mathematically?

The Delta Dirac distribution is defined as a limit of a sequence of functions, called the "approximating functions". These functions have a height of 1/ε at the point of interest and are 0 everywhere else. As the parameter ε approaches 0, the approximating functions converge to the Delta Dirac distribution.

What are some applications of the Delta Dirac distribution?

The Delta Dirac distribution has many applications in mathematics, physics, and engineering. It is commonly used to model point objects or point forces in physics, such as the point mass in mechanics or the point charge in electromagnetism. It is also useful in signal processing and image processing, where it can represent impulses or sharp changes in a signal.

What are the key properties of the Delta Dirac distribution?

The Delta Dirac distribution has several important properties, including: 1) it is even, meaning that δ(x) = δ(-x); 2) it is infinitely peaked at the origin, meaning that δ(x) = ∞ when x = 0; 3) it has unit area, meaning that its integral over the entire real number line is equal to 1; and 4) it satisfies the sifting property, meaning that ∫f(x)δ(x-a)dx = f(a) for any continuous function f(x).

How is the Delta Dirac distribution used in integration and differentiation?

The Delta Dirac distribution is often used in integration and differentiation as a tool to simplify calculations and solve certain types of problems. For example, it can be used to evaluate integrals with non-constant limits, or to solve differential equations with non-continuous coefficients. It can also be used to represent the derivative of a step function, which is not defined at the step point.

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