MHB Proving Parseval's Theorem for Schwartz Functions with Compact Support

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How to prove the following:

Suppose f is in the Schwartz Space ( smooth function with very fast decay). Its Fourier transform is smooth and has compact support contained in the interval (1/2,-1/2)

Show,

∫ (|f(x)|^2) dx = ∑ (|f(n)|^2) (where integral over R and sum up over n for all intergers)
 
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Sonifa said:
How to prove the following:

Suppose f is in the Schwartz Space ( smooth function with very fast decay). Its Fourier transform is smooth and has compact support contained in the interval (1/2,-1/2)

Show,

∫ (|f(x)|^2) dx = ∑ (|f(n)|^2) (where integral over R and sum up over n for all intergers)
I would start by using the Parseval theorem $$\int_{\mathbb{R}}|f(x)|^2dx = \int_{\mathbb{R}}|\hat{f}(\omega)|^2d\omega.$$

The sampling theorem (taking the sampling interval $\delta$ to be $\delta = 1$) says that $$f(x) = \sum_{n=-\infty}^\infty f(n)\phi(x-n),$$ where $\phi$ is a smooth function whose Fourier transform $\hat{\phi}$ is identically $1$ on $\bigl(-\frac12,\frac12\bigr)$ and has support in $(-\pi,\pi)$. Then $$\hat{f}(\omega) = \sum_{n=-\infty}^\infty f(n)\hat{\phi}(\omega -n)$$. Can you use that to show that $$\int_{\mathbb{R}}|\hat{f}(\omega)|^2d\omega = \sum_{n=-\infty}^\infty |f(n)|^2$$?

Edit. Correction: the Fourier transform of $\phi(x-n)$ is not $\hat{\phi}(\omega -n)$, but something like $e^{in\omega}\hat{\phi}(\omega) .$
 
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A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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