1)Another AHA. It is common knowledge that finding
coefficients of a harmonic representation, for a
periodic function, is much harder (mathematically)
than adding harmonics in order to reproduce the
original periodic function. This is because in the
first case one has to perform integration while in
the second case it is simply a matter of adding
several, often less than ten, harmonic components.
But in going from periodic functions (Fourrier series)
to non-repetitive pulses (Fourrier transforms) one
finds that the distribution of components is no longer
discrete. Reconstructing the original pulse from the
components also involves integration. The process
of finding the "inverse transform" (for example, to
verify that a solution is correct) is just as difficult as
the process of finding distributions of components
(finding the "direct transform.") Fortunately, computer-
based numerical integration can be used to solve
practical problems.
2) From my point of view difficulties are now mostly
conceptual, for example, "how to interpret negative
frequencies?" In the context of my illustrations
(replying to Roger) negative w has no physical
meaning; it appears because our very powerful
mathematical machinery is not limited to one kind
of problems. Is such statement acceptable?
Ludwik Kowalski