Distributions on non-test functions

In summary, a distribution is a map whose domain is a set of test functions. A defining quality of test functions is that they have compact support. If the function to be applied to is not of compact support, it is not a problem as long as you know what you are doing.
  • #1
pellman
684
5
The definitions of distributions that I have seen (for instance https://en.wikipedia.org/wiki/Distribution_(mathematics)#Distributions ) define a distribution as a map whose domain is a set of test functions. A defining quality of test functions is that they have compact support, which for most practical purposes means they are non-zero on only a finite interval (or intervals).

So why then are we comfortable writing something like ## \int^{\infty}_{-\infty} \delta(x-a) x^2 dx = a^2 ##? In the language of distributions this would be ## \delta_a(\phi) = a^2 ## where ## \phi(x) = x^2 ##. If we mean that ## \phi(x) = x^2 ## for all real x, is this a problem?

Is there an applied assumption here that we don't mean ## \phi(x) = x^2 ## on the entire real line but only on some compact subspace? If so, is this potentially a problem in physics which often deals with functions whose domain is explicitly infinite? I have never seen anyone applying the Dirac delta function have any concern that the function applied to is of compact support.
 
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  • #2
Not a problem as long as you know what you are doing. In example the integral is [itex]a^2[/itex] for all finite intervals containing the delta function, so passing to the limit is O.K.
 
  • #3
Lol. But I don't know what I'm doing.

I'm guessing that for the specific case of the delta distribution that it is fine to act on functions which are not of compact support even though distributions in general are only defined to act on test functions. Is this the case?
 
  • #4
pellman said:
A defining quality of test functions is that they have compact support, which for most practical purposes means they are non-zero on only a finite interval (or intervals).

The set of compact supported smooth functions is one class of test functions, denoted ##\mathcal{D}##. There are other classes including ##\mathcal{E}## which is the set of all smooth functions, and Dirac is defined as a functional of this space. Importantly, the distributions defined on ##\mathcal{E}## are characterized as those distributions with compact support (note I'm talking about the support of the distribution being compact, not the test function).
 
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Likes pellman and jasonRF
  • #5
Thanks, pwsnafu.
 

Related to Distributions on non-test functions

What is a distribution on non-test functions?

A distribution on non-test functions is a mathematical object that extends the concept of a function to include more general types of functions, such as those that are not differentiable or that do not have a well-defined integral. These distributions are often used in physics and engineering to model physical phenomena that cannot be described by traditional smooth functions.

How are distributions on non-test functions different from traditional functions?

Distributions on non-test functions are more general than traditional functions because they can include discontinuous, non-differentiable, or even singular functions. They are also defined in terms of integrals rather than pointwise evaluations, which allows them to handle a wider range of functions and phenomena.

What are some examples of distributions on non-test functions?

Some common examples of distributions on non-test functions include the Dirac delta function, the Heaviside step function, and the sign function. These are all functions that are not traditionally considered to be smooth or well-behaved, but can be defined as distributions and used in various mathematical and scientific applications.

How are distributions on non-test functions used in physics?

In physics, distributions on non-test functions are often used to model physical phenomena that involve point sources, such as particles or charges. The Dirac delta function, for example, is used to describe the concentration of mass or charge at a single point in space, while the Heaviside step function is used to represent abrupt changes in physical properties.

What are some properties of distributions on non-test functions?

Distributions on non-test functions have several important properties, including linearity, shift-invariance, and continuity. They also have a well-defined notion of differentiation and integration, which allows them to be used in a variety of mathematical operations. In addition, they have a rigorous mathematical framework that allows for the manipulation and analysis of these functions.

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