Could you please specify hilbert basis of $L^2([-1,1])$? How will be the representation of a function f $\in L^2([-1,1])$ by means of its Fourier series?
My solution:
$E_k=1/\sqrt2 e^{kit\pi}, k\in Z$
$f=\sum_{k \in Z} c_kE_k$
$c_k=<f,E_k> =\int_{-1}^1 f(t) \overline{E_k(t)}dt$
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Ali Ismayilov
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Do you know what a Fourier series and an hilbert basis are ? – Damien L Feb 04 '13 at 11:51
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Yes, I know. I would like to check that my solution is correct or not? – Ali Ismayilov Feb 04 '13 at 11:57
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"I would like to check my solution is correct or not" What is your solution? – Willie Wong Feb 04 '13 at 11:58
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I am editing now – Ali Ismayilov Feb 04 '13 at 11:59
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1Have you checked that the basis elements ($E_k$) are orthogonal to each other and are of unit norm? – Berci Feb 04 '13 at 12:58
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No, I have not checked it. – Ali Ismayilov Feb 05 '13 at 12:09