When discretization is commutative with dualization, then, under appropriate conditions, Pontryagin's minimum principle emerges as a consequence of the convergence of the discretization.
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More generally, given an abelian topological group G with Pontryagin dual G^, Parseval's theorem says the Pontryagin–Fourier transform is a unitary operator between Hilbert spaces L2(G) and L2(G^) (with integration being against the appropriately scaled Haar measures on the two groups.) When G is the unit circle T, G^ is the integers and this is the case discussed above.