The Schwartz space and the Fourier transform

Jordan Bell
August 17, 2015

1 Schwartz functions

Let 𝒮⁢(ℝn) be the collection of Schwartz functions ℝn→ℂ. For p≥0 and ϕ∈𝒮, write

∥ϕ∥p2=∑|ν|≤p∫ℝn(1+|x|2)p⁢|(Dν⁢ϕ)⁢(x)|2⁢𝑑x.

With the metric

d⁢(ϕ,ψ)=∑p≥02-p⁢∥ϕ-ψ∥p1+∥ϕ-ψ∥p,

𝒮 is a Fréchet space.

For a multi-index α and for ϕ∈𝒮, x↦xα⁢ϕ⁢(x) belongs to 𝒮 and we define Xα:𝒮→𝒮 by (Xα⁢ϕ)⁢(x)=xα⁢ϕ⁢(x). Dα⁢ϕ∈𝒮 and

∥Dα⁢ϕ∥p2=∑|ν|≤p∫ℝ(1+|x|2)p⁢|(Dν+α⁢ϕ)⁢(x)|2⁢𝑑x≤∥ϕ∥p+|α|2.

Because |{μ:|μ|=k}|=(n+k-1k),11 1 Arthur T. Benjamin and Jennifer J. Quinn, Proofs that Really Count: The Art of Combinatorial Proof, p. 71, Identity 143 and p. 74, Identity 149.

|{μ:μ≤ν}|≤|{μ:|μ|≤|ν|}|≤(n+|ν||ν|).

The product rule states

Dν⁢(f⁢g)=∑μ≤ν(νμ)⁢(Dμ⁢f)⁢(Dν-μ⁢g),

and with the Cauchy-Schwarz inequality we obtain for |ν|≤p,

|Dν⁢(Xα⁢ϕ)|2 =|∑μ≤ν(νμ)⁢(Dμ⁢ϕ)⁢(Dν-μ⁢Xα)|2
≤(n+pp)⁢∑|μ|≤p(νμ)2⁢|Dμ⁢ϕ|2⁢|Dν-μ⁢Xα|2,

and with this

∥Xα⁢ϕ∥p2 =∑|ν|≤p∫ℝn(1+|x|2)p⁢|(Dν⁢(Xα⁢ϕ))⁢(x)|2⁢𝑑x
≤∑|ν|≤p∫ℝn(1+|x|2)p⁢(n+pp)⁢∑|μ|≤p(νμ)2⁢|Dμ⁢ϕ|2⁢|Dν-μ⁢Xα|2⁢d⁢x
≤Cp⁢∥ϕ∥p+|α|2.

For g,ϕ∈𝒮 we have g⁢ϕ∈𝒮, and using the product rule we get

∥g⁢ϕ∥p2≤Cp,g⁢∥ϕ∥p2.

Therefore,

ϕ↦Dα⁢ϕ,ϕ↦Xα⁢ϕ,ϕ↦g⁢ϕ

are continuous linear maps 𝒮→𝒮.

2 Tempered distributions

For u:𝒮→ℂ, we write

⟨ϕ,u⟩=u⁢(ϕ).

𝒮′ denotes the dual space of 𝒮, and the elements of 𝒮′ are called tempered distributions. We assign 𝒮′ the weak-* topology, the coarsest topology on 𝒮′ such that for each ϕ∈𝒮 the map u↦⟨ϕ,u⟩ is continuous 𝒮′→ℂ.

For ψ∈𝒮, we define Λψ:𝒮→ℂ by

⟨ϕ,Λψ⟩=∫ℝnϕ⁢(x)⁢ψ⁢(x)⁢𝑑x,ϕ∈𝒮,

and by the Cauchy-Schwarz inequality,

|⟨ϕ,Λψ⟩|≤(∫ℝn|ϕ⁢(x)|2⁢𝑑x)1/2⁢(∫ℝn|ψ⁢(x)|2⁢𝑑x)1/2=∥ψ∥0⁢∥ϕ∥0,

whence Λψ∈𝒮′. It is apparent that ψ↦Λψ is linear. Suppose that ψi→ψ in 𝒮, and let ϕ∈𝒮. Then

|⟨ϕ,Λψi⟩-⟨ϕ,Λψ⟩|=|⟨ϕ,Λψi-ψ⟩|≤∥ψi-ψ∥0⁢∥ϕ∥0→0,

which shows that ψ↦Λψ is continuous. If Λψ=0, then in particular Λψ⁢ψ¯=0, i.e. ∫ℝn|ψ⁢(x)|2⁢𝑑x=0, which implies that ψ⁢(x)=0 for almost all x and because ψ is continuous, ψ=0. Therefore, ψ↦Λψ is a continuous linear injection 𝒮→𝒮′. It can be proved that Λ⁢(𝒮) is dense in 𝒮′.22 2 Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, volume I: Functional Analysis, revised and enlarged edition, p. 144, Corollary 1 to Theorem V.14.

For a multi-index α and u∈𝒮′, we define Dα⁢u:𝒮→ℂ by

⟨ϕ,Dα⁢u⟩=(-1)|α|⁢⟨Dα⁢ϕ,u⟩,ϕ∈𝒮.

For ϕi→ϕ in 𝒮, because Dα:𝒮→𝒮 and u:𝒮→ℂ are continuous,

⟨ϕi,Dα⁢u⟩=(-1)|α|⁢⟨Dα⁢ϕi,u⟩→(-1)|α|⁢⟨Dα⁢ϕ,u⟩=⟨ϕ,Dα⁢u⟩,

and therefore Dα⁢u∈𝒮′.

We define Xα⁢u:𝒮→ℂ by

⟨ϕ,Xα⁢u⟩=⟨Xα⁢ϕ,u⟩,ϕ∈𝒮.

For ϕi→ϕ in 𝒮,

⟨ϕi,Xα⁢u⟩=⟨Xα⁢ϕi,u⟩→⟨Xα⁢ϕ,u⟩=⟨ϕ,Xα⁢u⟩,

and therefore Xα⁢u∈𝒮′.

For g∈𝒮, we define g⁢u:𝒮→ℂ by

⟨ϕ,g⁢u⟩=⟨g⁢ϕ,u⟩,ϕ∈𝒮.

For ϕi→ϕ in 𝒮,

⟨ϕi,g⁢u⟩=⟨g⁢ϕi,u⟩→⟨g⁢ϕ,u⟩=⟨ϕ,g⁢u⟩,

and therefore g⁢u∈𝒮′.

For ψ∈𝒮, integrating by parts yields

⟨ϕ,Dα⁢Λψ⟩ =(-1)|α|⁢⟨Dα⁢ϕ,Λψ⟩
=(-1)|α|⁢∫ℝn(Dα⁢ϕ)⁢(x)⁢ψ⁢(x)⁢𝑑x
=∫ℝnϕ⁢(x)⁢(Dα⁢ψ)⁢(x)⁢𝑑x
=⟨ϕ,ΛDα⁢ψ⟩,

which implies that Dα⁢Λψ=ΛDα⁢ψ.

⟨ϕ,Xα⁢Λψ⟩=⟨Xα⁢ϕ,Λψ⟩=∫ℝnxα⁢ϕ⁢(x)⁢ψ⁢(x)⁢𝑑x=⟨ϕ,ΛXα⁢ψ⟩,

which implies that Xα⁢Λψ=ΛXα⁢ψ.

⟨ϕ,g⁢Λψ⟩=⟨g⁢ϕ,Λψ⟩=∫ℝng⁢(x)⁢ϕ⁢(x)⁢ψ⁢(x)⁢𝑑x=⟨ϕ,Λg⁢ψ⟩,

which implies that g⁢Λψ=Λg⁢ψ.

Because ϕ↦Dα⁢ϕ, ϕ↦Xα⁢ϕ, and ϕ↦g⁢ϕ are continuous linear maps 𝒮→𝒮 and because Λ:𝒮→𝒮′ is a continuous linear map with dense image, using the above it is proved that

u↦Dα⁢u,u↦Xα⁢u,u↦g⁢u

are continuous linear maps 𝒮′→𝒮′.33 3 Richard Melrose, Introduction to Microlocal Analysis, http://math.mit.edu/~rbm/iml/Chapter1.pdf, p. 17.

3 The Fourier transform

For Borel measurable functions f,g:ℝn→ℂ, for those x for which the integral exists we write

(f*g)⁢(x)=∫ℝnf⁢(x-y)⁢g⁢(y)⁢𝑑y=∫ℝnf⁢(y)⁢g⁢(x-y)⁢𝑑y,x∈ℝn,

and for those Borel measurable f,g:ℝn→ℂ for which the integral exists we write

⟨f,g⟩L2=∫ℝnf⁢(x)⁢g⁢(x)¯⁢𝑑x.

For ξ∈ℝn we define

eξ⁢(x)=e2⁢π⁢i⁢ξ⋅x,x∈ℝn,

and for ϕ∈𝒮 we calculate, integrating by parts,

(Dα⁢ϕ)*eξ=(2⁢π⁢i⁢ξ)α⁢ϕ*eξ.

We define ℱ⁢ϕ:ℝn→ℂ by

(ℱ⁢ϕ)⁢(ξ)=⟨ϕ,eξ⟩L2=∫ℝnϕ⁢(x)⁢eξ⁢(x)¯⁢𝑑x=∫ℝne-2⁢π⁢i⁢x⋅ξ⁢ϕ⁢(x)⁢𝑑x,ξ∈ℝn,

which we can write as

(ϕ*eξ)⁢(0)=∫ℝnϕ⁢(y)⁢eξ⁢(-y)⁢𝑑y=∫ℝnϕ⁢(y)⁢eξ⁢(y)¯⁢𝑑y=(ℱ⁢ϕ)⁢(ξ).

By Fubini’s theorem,

ℱ⁢(ϕ*ψ)⁢(ξ) =∫ℝnψ⁢(y)⁢(∫ℝnϕ⁢(x-y)⁢eξ⁢(x)¯⁢𝑑x)⁢𝑑y
=∫ℝnψ⁢(y)⁢(∫ℝnϕ⁢(x)⁢eξ⁢(x+y)¯⁢𝑑x)⁢𝑑y,

whence

ℱ⁢(ϕ*ψ)=(ℱ⁢ϕ)⁢(ℱ⁢ψ).

We calculate

ℱ⁢(Dα⁢ϕ)⁢(ξ)=((Dα⁢ϕ)*eξ)⁢(0)=((2⁢π⁢i⁢ξ)α⁢ϕ*eξ)⁢(0)=(2⁢π⁢i⁢ξ)α⁢(ℱ⁢ϕ)⁢(ξ),

whence

ℱ⁢(Dα⁢ϕ)=(2⁢π⁢i)|α|⁢Xα⁢ℱ⁢ϕ.

It follows from the dominated convergence theorem

(Dα⁢ℱ⁢ϕ)⁢(ξ) =∫ℝn(-2⁢π⁢i⁢x)α⁢e-2⁢π⁢i⁢x⋅ξ⁢ϕ⁢(x)⁢𝑑x
=(-2⁢π⁢i)|α|⁢∫ℝne-2⁢π⁢i⁢x⋅ξ⁢xα⁢ϕ⁢(x)⁢𝑑x
=(-2⁢π⁢i)|α|⁢ℱ⁢(Xα⁢ϕ)⁢(ξ).

Therefore

ℱ⁢Dα=(2⁢π⁢i)|α|⁢Xα⁢ℱ,Dα⁢ℱ=(-2⁢π⁢i)|α|⁢ℱ⁢Xα. (1)

Using the multinomial theorem,

(1+|ξ|2)p⁢|(Dν⁢ℱ⁢ϕ)⁢(ξ)|2 =∑k=0p(pk)⁢|ξ|2⁢k⁢|(Dν⁢ℱ⁢ϕ)⁢(ξ)|2
=∑k=0p(pk)⁢∑|α|=k(kα)⁢ξ2⁢α⁢|(Dν⁢ℱ⁢ϕ)⁢(ξ)|2
=∑k=0p(pk)⁢∑|α|=k(kα)⁢|(ξα⁢Dν⁢ℱ⁢ϕ)⁢(ξ)|2.

Applying (1),

|(ξα⁢Dν⁢ℱ⁢ϕ)⁢(ξ)|=(2⁢π)|ν|⁢(2⁢π)-|α|⁢|(ℱ⁢Dα⁢Xν⁢ϕ)⁢(ξ)|.

Then

∥ℱ⁢ϕ∥p2 =∑|ν|≤p∫ℝn(1+|ξ|2)p⁢|(Dν⁢ℱ⁢ϕ)⁢(ξ)|2⁢𝑑ξ
=∑|ν|≤p∫ℝn∑k=0p(pk)⁢∑|α|=k(kα)⁢|(ξα⁢Dν⁢ℱ⁢ϕ)⁢(ξ)|2⁢d⁢ξ
=∑|ν|≤p(2⁢π)2⁢|ν|⁢∑k=0p(pk)⁢(2⁢π)-2⁢k⁢∑|α|=k(kα)⁢∫ℝn|(ℱ⁢Dα⁢Xν⁢ϕ)⁢(ξ)|2⁢𝑑ξ.

Applying the Plancherel theorem, the product rule, and the Cauchy-Schwarz inequality yields

∫ℝn|(ℱ⁢Dα⁢Xν⁢ϕ)⁢(ξ)|2⁢𝑑ξ =∫ℝn|(Dα⁢Xν⁢ϕ)⁢(ξ)|2⁢𝑑ξ
=∫ℝn|∑β≤α(Dβ⁢Xν)⁢(Dα-β⁢ϕ)|2⁢𝑑ξ
≤∫ℝn∑β≤α|(Dβ⁢Xν)⁢(ξ)|2⋅∑β≤α|(Dα-β⁢ϕ)⁢(ξ)|2.

This yields

∥ℱ⁢ϕ∥p≤Cp⁢∥ϕ∥p,

whence ℱ:𝒮→𝒮 is continuous.

For p>n/2, using the Cauchy-Schwarz inequality and spherical coordinates44 4 http://individual.utoronto.ca/jordanbell/notes/sphericalmeasure.pdf we calculate

|(ℱ⁢ϕ)⁢(ξ)| ≤∫ℝn(1+|x|2)-p/2⁢(1+|x|2)p/2⁢|ϕ⁢(x)|⁢𝑑x
≤(∫ℝn(1+|x|2)-p⁢𝑑x)1/2⁢(∫ℝn(1+|x|2)p⁢|ϕ⁢(x)|2⁢𝑑x)1/2
=(∫0∞∫Sn-1(1+r2)-p⁢𝑑σ⁢rn-1⁢𝑑r)1/2⁢(∫ℝn(1+|x|2)p⁢|ϕ⁢(x)|2⁢𝑑x)1/2
=(πn/2⁢Γ⁢(p-n2)Γ⁢(p))1/2⁢(∫ℝn(1+|x|2)p⁢|ϕ⁢(x)|2⁢𝑑x)1/2
≤(πn/2⁢Γ⁢(p-n2)Γ⁢(p))1/2⁢∥ϕ∥p.