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Plancherel theorem

Statement

Assume countable choice and let $n \geq 1$ . Fourier transformation on Schwartz space extends uniquely to a surjective complex-linear isometry $F_{2} : L^{2} (R^{n}; C) \to L^{2} (R^{n}; C)$ . It preserves the first-variable-linear inner product, and hence is unitary.

Facts & Assumptions

Given: An integer $n \geq 1$ , The Axiom of Countable Choice ( ${A C}_{ω}$ ), and almost-everywhere classes as in The space $L^{p} (μ)$ as the quotient by null functions.

[F1]

Schwartz Parseval preserves pairings and norms (Parseval pairing on Schwartz space).

[F2]

Schwartz classes are dense in complex $L^{2}$ (Schwartz space is dense in L2).

[F3]

Fourier is onto Schwartz space (Fourier transform is a topological automorphism of Schwartz space).

[F4]

A bounded linear map on a dense subspace into a Banach space extends uniquely under countable choice (A bounded linear map from a dense normed subspace into a Banach space extends uniquely with the same norm).

[F5]

Complex $L^{2}$ is complete with the stated pairing and Cauchy–Schwarz (Complex completeness, density, and inner product: the consumer interface).

Proof

technique · direct

1.1F1F2F4F5given

View Schwartz functions as a normed subspace of $L^{2}$ : a continuous function vanishing a.e. vanishes everywhere, since any nonzero value persists on a positive-volume box. Thus the association with its class is injective. By [F1], Fourier is a bounded linear isometry on this dense subspace, and [F5] makes the target Banach. [F4] and [F2] give a unique bounded linear extension $F_{2}$ . For each fixed $f$ , countable choice selects a sequence $u_{k} \in S$ with $∥ u_{k} - f ∥_{2} < 1 / (k + 1)$ for $k \in N$ ; its Fourier images converge to $F_{2} f$ . The isometry on the subspace gives $∥ F_{2} f ∥_{2} = \lim_{k} ∥ {\hat{u}}_{k} ∥_{2} = \lim_{k} ∥ u_{k} ∥_{2} = ∥ f ∥_{2}$ . Independence of the sequence follows also from $∥ {\hat{u}}_{k} - {\hat{v}}_{k} ∥_{2} = ∥ u_{k} - v_{k} ∥_{2} \to 0$ .

2.1step 1.1F1F2F3F5given

For any $g \in L^{2}$ , [F2] and countable choice give $v_{k} \in S$ tending to $g$ . By [F3], define the uniquely determined $u_{k} = F^{- 1} v_{k} \in S$ . By [F1], $∥ u_{k} - u_{l} ∥_{2} = ∥ v_{k} - v_{l} ∥_{2}$ , so [F5] gives a limit $u \in L^{2}$ . Continuity of the extension in step 1.1 gives $F_{2} u = \lim_{k} v_{k} = g$ . Thus the extension is surjective.

3.1step 1.1step 2.1F1F5∎

For approximants $u_{k} \to f$ , $v_{k} \to g$ , Cauchy–Schwarz in [F5] bounds $∣ ⟨ u_{k}, v_{k} ⟩ - ⟨ f, g ⟩ ∣$ by $∥ u_{k} - f ∥_{2} ∥ v_{k} ∥_{2} + ∥ f ∥_{2} ∥ v_{k} - g ∥_{2} \to 0$ , since a convergent sequence is norm bounded. Apply the same estimate to their transform images and pass to the limit in [F1]. This proves pairing preservation. Steps 1.1 and 2.1 give the remaining unitary properties. Countable choice was used only in the cited interfaces and to select countable approximation sequences for each fixed input; no simultaneous arbitrary-index selection or Hilbert basis is needed.

Depends on

Dependency tree · 2 levels

Sources

Gerald Teschl, Topics in Real and Functional Analysis (2017)