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Sometimes it is useful to know less mathematics...

Perhaps the most simple example of an analytic function $f$, other than $0$, satisfying all three equalities is $f(x) = \sin(\pi x)/ (\pi x)$. (The specific constant $\pi$ plays an essential role in this result.) This is verified immediately by a change of variable $x'=\pi x$, giving $\int_{ - \infty }^\infty {f(x)\,{\rm d}x} = \int_{ - \infty }^\infty {f^2 (x)\,{\rm d}x} = 1 = \sum\nolimits_n {f(n)} = \sum\nolimits_n {f^2 (n)}$.

EDIT: TCL provided a (very interesting) generalization.

2 added 31 characters in body

Sometimes it is useful to know less mathematics... Perhaps the most simple example of a an analytic function $f$, other than $0$, satisfying all three equalities is $f(x) = \sin(\pi x)/ (\pi x)$. (The specific constant $\pi$ plays an essential role in this result.) This is verified immediately by a change of variable $x'=\pi x$, giving $\int_{ - \infty }^\infty {f(x)\,{\rm d}x} = \int_{ - \infty }^\infty {f^2 (x)\,{\rm d}x} = 1 = \sum\nolimits_n {f(n)} = \sum\nolimits_n {f^2 (n)}$.

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Sometimes it is useful to know less mathematics... Perhaps the most simple example of a function $f$ satisfying all three equalities is $f(x) = \sin(\pi x)/ (\pi x)$. (The specific constant $\pi$ plays an essential role in this result.) This is verified immediately by a change of variable $x'=\pi x$, giving $\int_{ - \infty }^\infty {f(x)\,{\rm d}x} = \int_{ - \infty }^\infty {f^2 (x)\,{\rm d}x} = 1 = \sum\nolimits_n {f(n)} = \sum\nolimits_n {f^2 (n)}$.