"Richard Hennessy" <rich.hennessy at verizon.net> writes:
> Hi,
> I have always thought that the main use of pwdelta() was to
> introduce a discontinuity in the ant-derivative of a function, then I
> found out about the derivatives of the pwdelta() function and the
> associated formulas and thought that their main use was to introduce
> discontinuities in the higher order anti-derivatives. Now I find out
> that this comes at a price. If you have a simple polynomial like
> x^2+x+1 and you want to put a discontinuity into its anti-derivative,
> then you can write (x^2+x+1)*pwdelta(x-2), but now the derivatives of
> the function do not stop at zero. Simple polynomials have a finite
> number of anti-derivatives and then you stop at zero. Not true for
> the modified polynomial with the pwdelta(). It has an infinite number
> of anti-derivatives and they are not trivial. The represent real
> curves on the Cartesian Coordinate system. I don?t pretend to
> understand why but this formula predicts this fact. If it is a fact,
> or is there a flaw in my reasoning?
> Rich
Well, if you start with p(x) * delta(x-k) where p is a polynomial of
degree n then, after differentiating n+1 times you end up with a
functional f. What is f? Firstly, you notice that:
When restricted to any domain not including k, f is zero
This is just from the fact that differentiation has a local definition
and x ? delta(x-k) is equal to the zero function away from k.
f(x) = p(k)*diff(delta(x-k), x, n+1)
To see that, consider the following chain of equalities (in
pseudo-Maxima notation)
integrate (phi(x)*diff(p(x)*delta(x-k), x, n+1), x) =
integrate(-diff(phi(x), x)*diff(p(x)*delta(x-k), x, n), x) = ... =
integrate((-1)^(n+1)*diff(phi(x), x, n+1)*p(x)*delta(x-k), x) =
(-1)^(n+1)*at(diff(phi(x), x, n+1), x=k)*p(k)
assuming that phi has compact support or something to get rid of the
boundary terms.
So the point is that multiplying by delta doesn't really make your
function(al) much more complicated: the polynomial disappears as
expected, just leaving the single atom at zero.
Note: I'm trained as an algebraic topologist, not a functional analyst,
so there might be some utter howlers in the above. I don't think
so though...
Rupert
-------------- next part --------------
A non-text attachment was scrubbed...
Name: not available
Type: application/pgp-signature
Size: 315 bytes
Desc: not available
URL: <http://www.math.utexas.edu/pipermail/maxima/attachments/20130719/75f0b5a2/attachment.pgp>;