Suppose I have a function f(x) which is a
rectangular pulse between x=2 and x=5,
[say f(x)=7] and zero everywhere else. I want
to find A(k), the Fourrier transform of that f(x).
The formula tells me that for any chosen k
A(k)=Integral of f(x)*cos(kx)*dx
In general the limits of integration are minus
and plus infinity but in practice they refer
to a region in which f(x) is not zero. Thus in
my example xmin=2 and xmax=5. During
the integration k remains a constant.
According to illustrations seen in books
A(k) is a functions which wiggles between
positive and negative values, depending
on k. In this message I was going to ask
how to justify negative A(k) on the intuitive
basis. But the answer came to me right now.
So instead of explaining why I was puzzled
let me share the qualitative explanation;
perhaps it can help somebody.
The function I am integrating wiggles between
-7 and +7. The number of wiggles (cycles)
is different for each k. If k=6, for example, then
the integral is zero; the number of + swings and
the number of - swings are even. But for k equal
to 6.1, for example, the integral is positive because
the added positive area (under the function and x
axis) does not have a negative counterpart. For
k=5.9, on the other hand, the integral becomes
negative because the last positive swing is not
completed and the sum of negative areas is still
larger than the sum of positive areas.
It is really an AHA moment for me; the nonzero
values of A(k) come from situations in which the
cos(k*x) function does not fit the region of x in
which f(x) is not zero. A small change in the width
of my f(x) pulse can change the shape of the A(k)
pulse significantly.
A connection with the uncertainty principle is
sometimes made in chapters describing Fourrier
transforms. The A(k) becomes wider when f(x)
become narrower, and vice versa.
Ludwik Kowalski