Re: Fourier Transforms in general settings


I'm keen to get a better intuition as to how Fourier
transforms
are defined and understood to work in general
settings, such
as random graphs and even groups, and the uses to
which
they can be put in these.

In other words, what is a Fourier transform in these
situations
and are there any other types of involution (if that
is the correct
word) with similar properties?

I think you'd want to take a look at this:

http://en.wikipedia.org/wiki/Pontryagin_duality


I gather a "classical" FT of a function can be
understood as the
function being a point in an infinite dimensional
space, with the
point projected onto an orthogonal frame. That makes
sense
on account of the well-known property of being able
to multiply
the function by a suitable factor and integrate away
all but one
term, analogous (I suppose) to taking the scalar
product of
a vector by one of the basis elements.

But in that case how is the FT unique? Or is the
answer that
there's an infinite number of possible orthogonal
bases, such
as Chebychev [sp?] series and so forth, and an FT is
merely
one?

As always feel free, if necessary, to change the
question to
one which makes more sense, or answer only part,
Defining
FTs for groups or graphs would be a good start - All
my
web searches on these have landed on
subscription-only
journals, which is very tiresome!

Thanks in anticipation.
.

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