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Let $L=1, A=1$ and $f\in L^2([-L/2, L/2])$, with Fourier series $f^{t}=\sum_{n=-K}^{K}a_n \exp(2j\pi xn/L),$ truncated at $K$. Has this function, $f^{t}$, any relation with Fourier Inverse Transform of $\hat{W}(w)*\hat{f}(w)$?.

I begin calculated $f^t$ and Inverse Fourier of $\hat{W}(w)*\hat{f}(w)$ I get: $\sum_{n=-K}^{K}\dfrac{A}{2\pi j n}(1-\exp(-\pi j n x))\exp(2\pi j n x)$

and

$\int_{-K}^{K}\dfrac{A}{2\pi j w}(1-\exp(-\pi j w x))\exp(2\pi j w x)dx$ respectively, then I think that relation is when $K\rightarrow \infty$. The first and second expression are equals, Is true this?

Thanks for your replies.

pdta: $W(x) = \dfrac{\sin(2\pi Kx)}{\pi x}\;$ and $f(x) = 0,$ if $-L/2 \leq x < 0$; $f(x) = A,$ if $0 \leq x < L/2$

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    I edit again a question, sorry2012-10-30

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