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This is the precise meaning of zero away from the reciprocal lattice. A finite specimen instead has broadened reciprocal-lattice peaks of a finite crystal; exact delta-function support is an infinite-periodicity idealization.

Ψ

c_{n} = \frac{1}{2 π} ⟨ u, Ψ (x) e^{- in x} ⟩, n \in Z .

2 π

. The finite Schwartz seminorm bound for

u

, restricted to the fixed compact support of

Ψ

, gives

∣ c_{n} ∣ \leq C (1 + ∣ n ∣)^{N}

for some

C, N

. Indeed, only finitely many derivatives of

Ψ e^{- in x}

occur, and each contributes at most a fixed power of

∣ n ∣

ϕ

, the preceding Fourier series formula and

ϕ (n) = 2 π ϕ (- n)

yield

P ϕ (x) = \sum_{n \in Z} ϕ (- n) e^{in x} .

All derivatives of this series converge uniformly. The localization formula in the previous answer therefore gives

⟨ u, ϕ ⟩ = ⟨ u, ϕ ⟩ = ⟨ u, Ψ P ϕ ⟩ = n \sum ϕ (- n) ⟨ u, Ψ e^{in x} ⟩ = 2 π n \sum c_{n} ϕ (n) .

u = 2 π n \in Z \sum c_{n} δ_{n} in S^{'} (R) .

The polynomial bound on

c_{n}

and rapid decay of

ϕ (n)

make every displayed pairing absolutely convergent. In particular there are no derivatives of Dirac delta distributions in this formula.

δ_{n}

(2 π)^{- 1} e^{in x}

u = n \in Z \sum c_{n} e^{in x} in S^{'} (R) .

Explicitly, for every Schwartz function

θ

, the pairing of the partial sum with

θ

\sum_{∣ n ∣ \leq M} c_{n} θ (- n)

, which converges absolutely to

⟨ u, θ ⟩

. The Fourier transform formula also proves that

c_{n}

is independent of the chosen smooth periodic partition of unity. For a locally integrable

2 π

periodic function it reduces, by splitting the integral into periods, to

c_{n} = (2 π)^{- 1} \int_{0}^{2 π} u (x) e^{- in x} d x