Re: Beginner questions on distributions
- From: David C. Ullrich <dullrich@xxxxxxxxxxx>
- Date: Sat, 28 Feb 2009 09:10:51 -0600
On Fri, 27 Feb 2009 14:10:10 -0800 (PST), "Achava Nakhash, the Loving
Snake" <achava@xxxxxxxxxxx> wrote:
Hi Group,
On and off, I have been trying to pick up some knowledge of EE. One
of the first weirdities they hit you with is the idea of the unit
impulse function, aka the Dirac delta function which is has the
properties that delta(0) = 1, d(x) = 0 for x not equal to 1, and the
integral form minus infinity to infinity of delta(x) dx = 1.
No, for the delta you're talking about it's certainly not true
that delta(0) = 1. delta is not a function at all.
What it really is is a linear functional, exactly as you
suggest below, and "int f delta dx" is just unfortunate
notation for delta(f), which is often written <f, delta>,
and which is simply _defined_ to be f(0).
When I
first learned of it I concluded that Electrical Engineers and
Physicists were all insane.
I later mellowed in my views (at least as they pertain to the unit
impulse function) as it is on the same level of whackiness as idea of
a point mass, which I have always managed to swallow whole without
gagging, and in fact, it even took me a while to realize that there
was an actual problem here.
Anyway, I have been given to understand that Distributions, aka
Generalized Functions, come to the rescue as far as the unit impulse
and other so-called singularity functions are concerned.
As I understand it, a distribution D is actually a bounded linear
functional on the space of all C^infinity functions from reals to
reals with compact support. The analog of integrating D times a
regular function from minus inf to inf is then given as D(g), or some
appropriate limit if the g doesn't happen to be of the above type. I
don't actually know what this appropriate limit might be, but I can at
least guess.
Now EE types have a compulsion to take convolutions and to do Fourier
analysis with all the functions that they use, and so it is going to
be necessary to do that with distributions as well, not to mention
differentiating. Other arithmetic operations are a little bit more
obvious. It seems to me that, in order to do these things, we need
the following definitions: What is the integral of D times g(x) from
a to b where a and b are arbitrary real numbers or plus or minus
infinitsy? What is the analog of the integral for D(t - x) times g(x)
over any interval, or even only over the interval from minus infinity
to infinity? What is the derivative of D? I can play a game with
integration by parts to come to the conclusion that the derivative of
D takes g to the value you get when D is applied to the derivative of
g. Is that correct? What else do I need to know how to do in order
to be able to do the usual analytic tricks using distributions instead
of functions? I only really care about going from R to R at the
moment .
Now I realize that I can probably find the answers to these questions
in a variety of textbooks, and I do want the mathematician version of
othis so that I can deal better with the EE version of this, but my
analysis background is fairly weak, and I simply don't have the time
to learn enough to make this a viable option. So please help.
If you don't have time to learn the subject then you don't
have time to learn it. It's not all that hard but there are a lot
of fussy details - if you actually want the mathematician
version you're not going to get it fron a few usenet posts.
Thanks,
Achava
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.
- References:
- Beginner questions on distributions
- From: Achava Nakhash, the Loving Snake
- Beginner questions on distributions
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