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I have just started reading some things about generalised functions, and i have some question. The source I am reading from is a book of partial differential equations so it's not a very formal introduction to generalized functions and functionals, but there are some basic things to help understand the Green function etc...

First of all, it says that a functional of an integrable on any closed interval function f is a mapping between a set of continuous and infinite times derivable functions (h) and the real numbers. The mapping is defined by the following rule:

[tex]D(h)=\int_{-\infty}^{\infty}f(x)h(x) dx[/tex]

Then, it goes one to define the generalised function as a linear and continuous functional. I will use the symbol <f,h> for the generalised function [tex]T(h)=<f,h>=\int_{-\infty}^{\infty}f(x)h(x) dx[/tex]

The derivative of T is defined like this: [tex]T^{(n)}(h)\equiv(-1)^{n} T(h^{(n)})[/tex]

and it is always a generalised function. But in the book, it also mentions that to reach this definition, we thought of the functional of <f',h> with f' integrable on any interval,

So, my basic question is(concerning delta, too): When we write <f,h> we mean a generalised function. So does this mean that f(x) exists and is a real function and we can write f(x), meaning the real function from which the functional occured? This actually relates to delta. I mean, delta is defined as the derivative of the functional of Heaviside. The Heaviside function (let's call it H(x)) is integrable on any interval [a,b] so we can define <H,h>=T(h). Its first derivative is T'(h)=delta(h). But H'(x) isn't integrable on any [a,b], a<0<b, right? So what do we mean by writing delta(x)??

I am afraid that my writing is too messed up for anyone to understand, so I will restate my question in another way and then end the topic. Is there any real integrable function f so that <f,h>=delta?? If there isn't any, how can we say that delta is a generalised function??

This is actually what troubles me also when I read delta(x)? How can we do the transition and define delta as a real function, too??

That's all. Sorry for the size of the post and my bad English.

Thanks

Note: I have checked arildno's post on delta. The theory mentioned there is a bit different, and I don't really have much time to read and understand what he writes, because an exam approaches, so I would appreciate if any answer was based on the definitions I mention.

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# Questions about generalised functions, and delta

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