Re: Is the delta function absolutely integrable?

On Mon, 27 Jul 2009 08:04:25 -0700 (PDT), vv <vanamali@xxxxxxxxxxx>
wrote:

Which ever way you slice, this delta function is monkey business,
really! Even the man on the street knows that int delta(t) dt = 1.

The man on the street knows a lot of things. If we're careful about
our definitions it's not true that the integral of the delta function
is 1. Because the "delta function" is not a function, it's a
distrubution, and distributions simply don't have integrals.

The question I have is, what about int |delta(t)| dt, i.e., the
integral of the absolute value of the delta function ?

This makes less sense - a distribution does not have an absolute
value.

I don't know what level you're at mathematically. In various
books on, for example, differential equations, the author
talks about the delta function as though it were a function
with an integral. But it's not so - they just talk that way
in those books because they assume the reader doesn't
have the mathematical background to follow the real
explanation.

There are
sequences that tend to delta that are absolutely integrable, but what
got me confused is that the sinc function (sin pi x / pi x) also tends
to the delta function as it grows skinnier and taller; BUT, this guy
is not absolutely integrable. So, what's the verdict on absolute
integrability of the Dirac delta? Thanks!

--VV

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.)
.

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