E2 Signal seeds and lattice addresses
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Let \(1\leq p\leq\infty\). For a countable set \(G\), write \(\ell^p(G)\) for the usual complex sequence space. For an integer \(d\geq1\), when \(G={\mathbb Z}^d\), translation by \(a\in{\mathbb Z}^d\) is
Let \(\delta_0\) be the impulse at the origin.
Theorem E2.1: Impulse generation
For \(1\leq p<\infty\),
The same statement holds in \(c_0({\mathbb Z}^d)\) with the supremum norm. In \(\ell^\infty({\mathbb Z}^d)\), the closed span of the translation orbit is exactly \(c_0({\mathbb Z}^d)\).
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Proof of Theorem E2.1
The translation orbit is the standard coordinate family \((\delta_a)_{a\in{\mathbb Z}^d}\). Its finite span is the space of finitely supported sequences. Finitely supported sequences are dense in \(\ell^p\) for finite \(p\) and in \(c_0\). Their closure in the supremum norm is \(c_0\), which is a proper closed subspace of \(\ell^\infty\).
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Remark E2.2
The distinction at \(p=\infty\) is structural. The impulse generates every finitely supported signal algebraically, but it is not a topological cyclic seed for all bounded signals.
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Proposition E2.3: Finite periodic signals
Let \(G={\mathbb Z}^d/L\), where \(L\leq{\mathbb Z}^d\) has finite index. The translation orbit of the impulse at the identity is a basis of \({\mathbb C}^G\). In particular, one impulse generates every signal on the finite periodic lattice.
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Proof of Proposition E2.3
Translation sends the impulse at the identity to the impulse at each element of \(G\). These impulses form the standard basis of \({\mathbb C}^G\).
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