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the continuous Fourier Transform is often defined on a finite interval, usually [tex][-\pi,\pi][/tex]:

[tex]\hat{f_k} = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-ikx}dx[/tex]

If I understood correctly, this allows the basis functions to be defined so that they have norm=1, and they form anorthonormal basisfor [tex]L^2([-\pi,\pi])[/tex].

Now, I get confused when one tries to compute the FT of a functionfin the whole [tex]\mathcal{R}[/tex] because:

1) The 2-norm of the basis functions goes to [tex]+\infty[/tex]

2) They are not square integrable

3) Should I conclude that the basis-functions are orthogonal but NOT orthonormal?

4) If (1,2,3) are correct, then what is the space spanned by the basis-functions?

5) Is it possible to define an orthonormal basis for [tex]L^2(\mathcal{R})[/tex] with Fourier basis ?

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# Fourier Basis functions question

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