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Weighted Fourier transforms of Schwartz functions are dense in L2

Statement

Assume Countable Choice. For every $n \geq 1$ and $s \in R$ , the set ${⟨ ξ ⟩^{s} F (u) (ξ) : u \in S (R^{n})}$ is dense in complex $L^{2} (R^{n})$ . In fact it contains every frequency function in $C_{c}^{\infty} (R^{n})$ .

Facts & Assumptions

Given: Countable Choice, $n \geq 1$ , and $s \in R$ .

[A1]

Countable Choice is the principle of selecting one element from each nonempty set in a countable family (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

Both bracket powers multiply Schwartz space continuously and are mutual inverses (Real powers of the Japanese bracket act on Schwartz space).

[F2]

Fourier transformation is onto Schwartz space and has a Schwartz-valued inverse (Fourier transform is a topological automorphism of Schwartz space).

[F3]

Under Countable Choice, complex $C_{c}^{\infty} (R^{n})$ is dense in Euclidean complex $L^{p}$ for every finite $p$ (Complex finite-simple and smooth compact-support density for finite p).

[F4]

The weight and transform in this claim are the ones used in the preceding candidate form (Weighted Fourier candidate norm on Schwartz space).

[F5]

Complex compactly supported smooth functions are defined componentwise, and their derivatives are componentwise (Complex Lp classes and Euclidean test-function conventions).

[F6]

Schwartz space consists of actual smooth functions with all polynomially weighted derivative seminorms finite (Schwartz space and its seminorms).

Proof

technique · Exact preimage construction followed by smooth density

1.1F1F4F5F6given

Fix an arbitrary $h \in C_{c}^{\infty} (R^{n}; C)$ . By [F5], its components and every derivative are continuous with compact support, so each $ξ^{α} \partial^{β} h$ is bounded and [F6] gives $h \in S$ ; [F1] then gives $q = ⟨ ξ ⟩^{- s} h \in S$ .

2.1A1F2step 1.1

By [F2], $u = F^{- 1} q$ belongs to Schwartz space, and pointwise $⟨ ξ ⟩^{s} F (u) = ⟨ ξ ⟩^{s} q = h$ . Thus every such $h$ is in the weighted Fourier image.

3.1A1F3step 2.1∎

For any $g \in L^{2}$ and $ε > 0$ , [F3] with $p = 2$ supplies $h \in C_{c}^{\infty}$ with $∥ g - h ∥_{2} < ε$ ; step 2.1 puts this same $h$ in the weighted Fourier image, proving that image dense in $L^{2}$ .

Depends on

Used by

The Bessel completion embeds canonically in tempered distributions Theorem

Dependency tree · two levels

37 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Semyon Dyatlov, Lecture Notes for 18.155, current revision (standard reference, not scraped)
Richard B. Melrose, Differential Analysis, Chapter 3 (standard reference, not scraped)