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- Thread starter Amok
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A distribution function F(x), however, is defined as F(x) = P(X[tex]\leq[/tex]x) - the probability that the random variable X takes a value less than or equal to x. So the Dirac delta is a distribution, not a distribution function. In fact, the distribution function of the Dirac delta is the Heaviside step function.

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I don't think I'm confusing these concepts. The distributions I'm talking about are functionals, not functions. And the Heaviside function is the ant-derivative of the dirac delta in the sense of distributions. I think the correct term for what I'm talking about is "temperate distribution".

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arildno

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https://www.physicsforums.com/showthread.php?t=73447

It is rather informal on the technical distribution concept, but more rigorous in establishing the connection between the Dirac function(al) and the integral representation of it.

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arildno

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A distribution is a linear functional, Amok.

There might be some other technicalities here, but that is essentially what it is.

Thus, a distribution D on some function space satisfies the condition of linearity, most importantly D(a*f+b*g)=a*D(f)+b*D(g), where f and g are any two functions in the function space, and a and b scalars.

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Consider functions on some interval as vectors in an abstract vector space. We can then consider linear functionals which are linear mappings from functions to numbers. These form the dual space. If a vector space is finite dimensional and has an inner product we can express every functional using the inner product with another vector:

[tex] f \mapsto \langle g,f\rangle[/tex]

The integral of the product of two functions over an interval forms a good inner product and we can express many of the linear functionals via:

[tex] f \mapsto \int_{x_1}^{x_2} g(x)f(x)dx[/tex]

But since the function space is infinite dimensional there are "more" dual vectors i.e. linear functionals than there are vectors i.e. functions. For example we cannot express the evaluation mapping:

[tex] f \mapsto f(0)[/tex]

as an integral of f with another function. So we invent a dummy function name to hold the place of that missing function in the integral notation. And that is the delta "function".

It allows us to continue using the integral inner product to express those linear functionals which are not actually expressible as integrals.

[tex] f \mapsto f(0)=\langle \delta , f\rangle = \int_{x_1}^{x_2} \delta(x)f(x)dx[/tex]

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