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Re: Fourier transforms



Regarding the assertion
>> that all values of |A(k)| would be positive, for all |k| less than
>> pi/halfwidth.

I wrote:
>> It's easy to construct nontrivial counterexamples ... unless you
>> put heavy restrictions on the shape of the pulse.

On 10/15/2003 12:52 PM, John Mallinckrodt wrote:

> Yes, but I was referring to Ludwik's VERY heavily restricted example
> in which case the integral amounts to integrating a cosine function
> over a symmetric interval. The first time that integral becomes
> negative is when the interval exceeds one period.

We agree |A(k)| is positive for small k for
Ludwik's example ... however
-- The symmetry of the pulse has got nothing to do with it.
-- I suppose that's good, because in fact the example that
Ludwik originally offered was not symmetric.
-- The fact that Ludwik's pulse is everywhere positive
has got quite a lot to do with it.
-- I picked on this point because the original assertion
did not carry any warnings as to the heavy restrictions
on its validity. I was worried that innocent readers
would take the assertion for a general principle,
which it most certainly is not.


Specific counterexample:
____
______ ______
__ __

Note that |A(0)|==0 even if this is symmetric
(origin in the middle) and even if it is not
symmetric (origin elsewhere). This contradicts
the assertion, no matter what you take as the
"width" of the pulse.