Re: Distibution and measure on an interval

From: A N Niel (anniel_at_nym.alias.net.invalid)
Date: 03/27/05 

Date: Sun, 27 Mar 2005 05:57:45 -0500

In article <1111912701.270455.247460@l41g2000cwc.googlegroups.com>,
<bryant_j_j@yahoo.com> wrote:

> hi,
>
> my knowledge of distributions (in particular generalized
> derivatives) is rather limited, so i was wondering about the following:
>
> 1. does the exist a distribution D and an associated (singular)
> measure M on the interval [-\pi,pi] such that for a function f which is
> differentiable everywhere on (-\pi,\pi):
>
> f'(a)=\int_{-\pi}^{\pi} f(x)D(x) dx
> =\int_{-\pi}^{\pi} f(x)dM(x)
>
> for all a \in [-\pi,\pi], where f'(x) is the derivative of x w.r.t to
> f?

No, differentiation at a is not a measure, it is the derivative
of a measure. There is such a distribution, essentially
the derivative of delta. That object is sometimes called
a "dipole" by mathematical physicists. It _is_ concentrated
at the single point a, though.

>
> 2. Will M have to be a non-negative valued measure?
>
> 3. References for any answer to the above.
>
> (hope my use of a distribution in \int_{-\pi}^{\pi} f(x)D(x) dx is
> correct).
>

Mathematical physicists are happy writing things that way.
Mathematicians may prefer something like < f , D > so show
the action of distribution D on test function f.

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