Fourier Basis functions question

[mnb96]

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

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

If I understood correctly, this allows the basis functions to be defined so that they have norm=1, and they form an orthonormal basis for $$L^2([-\pi,\pi])$$.
Now, I get confused when one tries to compute the FT of a function f in the whole $$\mathcal{R}$$ because:

1) The 2-norm of the basis functions goes to $$+\infty$$
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 $$L^2(\mathcal{R})$$ with Fourier basis ?

 

[bartek2009]

I'm not familiar with a 2-norm but the basis is normalised "in my book" like so:

$$\int_{-\pi}^{\pi}\phi_{k}^{*}\phi_{k}dx = \frac{1}{2\pi}\int_{-\pi}^{\pi}e^{+ikx}e^{-ikx}dx = \frac{1}{2\pi}\int_{-\pi}^{\pi}dx = 1$$

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[mnb96]

Maybe I used an incorrect term, but indeed, you proved that the squared norm of the basis-functions are always 1.
The important points of my question (1,2,3,4,5) arise when the bounds of the integral are not anymore $$[-\pi,\pi]$$ but they become $$(-\infty,+\infty)$$

Basically, I wanted you to consider:
$$\int_{-\infty}^{+\infty}\phi_{k}^{*}\phi_{k}dx$$
and then, answer those questions.

 

[bartek2009]

I see what you mean now. In this case, I think you cannot define the basis.
But maybe there is a way to expand functions in an unnormalised basis.

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