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Let $v(x)\in L^1\cap L^2$ be a function, and $w(y)\in L^2$ be its Fourier transform. Let $\{w_k\}$ be an infinite sequence of samples from $w(y)$, sampled $T$ apart. Now,

$v(x)=\sum_{k=-\infty}^{\infty}w_k e^{ikx}$

in the $L^2$ norm sense. Is it possible to write $w_k=\displaystyle\int_0^{2\pi}v(x)e^{-ikx}dx$, since $v(x)$ is Riemann integrable, or should I use the inverse discrete transform?

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    Ok, thanks. I guess my main doubt was whether the sequence can be recreated exactly from the inverse Fourier, given the space that the two functions are in. If you just copy what you've written here in the answer field, I'll accept it.2011-03-10

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The distance between the sampling points $T$ should be 1 for your equations to hold.

Assuming everything to be nice the discrete back transform goes over to a Riemann integral as $n\to\infty$. So you can use your formula to calculate the back transform...

As always: functions where everything goes well lie dense and the resulting $L^2$ norms are bounded by Plancherel such that this argument can be extended to arbitrary functions in $L^2$.

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    I guess if $T$ is not one, you should integrate $k$ up to $2\pi/T$.2011-03-10