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Inner Product : Fourier Transform

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Please see the attached file for the fully formatted problem.

2. Let p be a fixed and given square integrable function, i.e.
0 < S g(x)g(x) dx = ||g^2|| < inf
The function g must vanish as |x| ---> inf. Consequently, one can think of g as a function whose non-zero values are concentrated in a small set around the origin x = 0.
Consider the concomitant "windowed" Fourier transform on L2(?inf, inf) , the space of square integrable
functions,

Let h(w, t) be an element of the range space . It is evident that

is an inner product on
FIND a formula for (Tf1, Tf2) in terms of the inner product
(f1,f2) Ef f(x)f2(x) dx
on L2(?oo,oo).

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Solution Summary

A proof involving an inner product and a Fourier transform is provided. The solution is well presented.

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