"I'm definitely not an expert on this subject, but I wanted to check that
you'd heard of "weak derivatives" and "distributions" from functional
analysis."
Yes, I know about distributions and I am thinking of diracdelta as a limit
of very spiked distributions with an increasingly small variance. A
function class I have used is
assume(s>0);
f[0](x) := exp( -x^2/(2*s) ) / sqrt( 2 * s * %pi);
for i : 1 thru n do define( f[i](x), diff( f[i-1](x), x ));
as s approaches zero this is a way to double check results, but f[0](x) is
an even function and I was thinking that may not always have to be the case,
in general. This function works at least for polynomials but it is pretty
limited. You can't even do limit(integrate(f[0](x-a) * sin(x), x, minf,
inf), which means I should find another function.
limit(integrate(x^3*f[0](x-a),x,minf,inf),s,0)
> a^3
limit(integrate(x^3*f[1](x-a),x,minf,inf),s,0)
> -3*a^2
"For delta, we can do slightly better and, if we define delta as a limit
of functions in some space of distributions, I'm pretty certain that the
value of (1) is independent of the approximating functions you choose as
long as phi is continuous at zero. If phi is *not* continuous at zero,
however, the answer depends on your approximating sequence."
I have searched for other functions that may be used for this type of
analysis without finding many good substitutes. Going the abstract way and
defining a distribution class that is independent of the details of the
chosen function is rather difficult. I am pretty certain you are right that
it depends on the chosen distribution and if phi is discontinuous at zero
the answer will change depending on f[n](x).
Of course this makes doing math with diracdelta somewhat harder.
pwint(pwdelta(x-a)*between(x,0,a,closed),x,minf,inf) will be wrong unless I
can find a way to test for continuity at a. I have a function that can do
that but I think it needs work.
Rich
-----Original Message-----
From: Rupert Swarbrick
Sent: Sunday, April 14, 2013 5:29 PM
To: maxima at math.utexas.edu
Subject: Re: [Maxima] DiracDelta
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