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


With the metric


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


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.


The product rule states


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

|Dν⁢(Xα⁢ϕ)|2 =|∑μ≤ν(νμ)⁢(Dμ⁢ϕ)⁢(Dν-μ⁢Xα)|2

and with this

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

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




are continuous linear maps 𝒮→𝒮.

2 Tempered distributions

For u:𝒮→ℂ, we write


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


and by the Cauchy-Schwarz inequality,


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


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


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


and therefore Dα⁢u∈𝒮′.

We define Xα⁢u:𝒮→ℂ by


For ϕi→ϕ in 𝒮,


and therefore Xα⁢u∈𝒮′.

For g∈𝒮, we define g⁢u:𝒮→ℂ by


For ϕi→ϕ in 𝒮,


and therefore g⁢u∈𝒮′.

For ψ∈𝒮, integrating by parts yields

⟨ϕ,Dα⁢Λψ⟩ =(-1)|α|⁢⟨Dα⁢ϕ,Λψ⟩

which implies that Dα⁢Λψ=ΛDα⁢ψ.


which implies that Xα⁢Λψ=ΛXα⁢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


are continuous linear maps 𝒮′→𝒮′.33 3 Richard Melrose, Introduction to Microlocal Analysis,, p. 17.

3 The Fourier transform

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


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


For ξ∈ℝn we define


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


We define ℱ⁢ϕ:ℝn→ℂ by


which we can write as


By Fubini’s theorem,

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



We calculate




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


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

Applying (1),



∥ℱ⁢ϕ∥p2 =∑|ν|≤p∫ℝn(1+|ξ|2)p⁢|(Dν⁢ℱ⁢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

This yields


whence ℱ:𝒮→𝒮 is continuous.

For p>n/2, using the Cauchy-Schwarz inequality and spherical coordinates44 4 we calculate

|(ℱ⁢ϕ)⁢(ξ)| ≤∫ℝn(1+|x|2)-p/2⁢(1+|x|2)p/2⁢|ϕ⁢(x)|⁢𝑑x