"Richard Hennessy" <rich.hennessy at verizon.net> writes:
> My answer of 1/2 is based on assumptions that diracdelta approximating
> functions are even functions. It is not necessary for that to be the
> case. I think Maple is right and integrate(unit_step(x) *
> diracdelta(x), x, minf, inf) should be undefined. pw.mac get?s it
> wrong. Sorry for the noise, I know this subject is a headache.
>
> pwint(unit_step(x) * pwdelta(x),x,minf,inf) gives 1/2, should be
> ?und. There are more cases like this which are also wrong.
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. If you talk to a mathematician (or analyst, at least), the
only way to work out what the "right" value is for this sort of thing is
to use that framework.
In particular, you define delta(x) to be the weak derivative of
step(x). Now, I *think* that you're right and
integrate (delta(x)*step(x), x, -inf, inf)
is undefined, but that is because I'm pretty certain
integrate (delta(x)*phi(x), x, -inf, inf) (1)
is only defined by the theory of distributions when phi is continuous
(and step definitely isn't continuous!). I think the general theory says
that (1) should have a definition with delta being any distribution as
long as phi is smooth and decays sufficiently rapidly.
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. (If you've
seen Cauchy completions, this is exactly what's going on: we start with
functionals that look like g -> integrate(f * g) then take all limits of
Cauchy sequences of such things)
I looked quickly, and I can't find a helpful Wikipedia article. There
are some course notes from Stanford at
http://math.stanford.edu/~andras/220-4.pdf
which look good. They're not too scarily abstract and the first couple
of pages (prose!) seem to explain what's going on rather better than I
can.
Disclaimer: The above is my hazy recollections from analysis lectures
several years ago so may or may not have glaring errors...
Rupert
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