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Home , Fourier transform

Chapter 5

Properties of the transform

The purpose of this section is to raise our level of sophistication of the analysis of the Fourier transform, and to make up our backlog of analytic justification of our work in the previous several sections. The Fourier trans- form plays a very important role in analysis, and for this reason it has been thoroughly analyzed from many points of view, by many people in many different settings. We will work through the bodies of two of the principal settings in this subsection, the ones most important to analysis and PDE. The cases are

(1) Hilbert spaces: covered in Section 5.1.

(2) Schwartz class: covered in Section 5.2.

5.1. Hilbert spaces

Definition 5.1. The usual for us is L2(Rn), which consists of the square-integrable measurabe functions on Rn;

L2(Rn)= f(x) Lebesgue measurable in Rn, f(x) 2dx < + . { ZRn | | ∞}

Proposition 5.2 (Elementary properties of L2(Rn)). (1) The space of func- tions L2(Rn) is a linear space, namely if f,g L2, then this implies that ∈ αf + βg L2. ∈ 55 56 5. Properties of the Fourier transform

(2) The topology on L2(Rn) is given by a norm, which is homogenous of degree 1. 1/2 2 f L2 = f(x) dx k k ZRn | |  αf 2 = α f 2 , f 2 = 0 f = 0 . k kL | |k kL k kL ⇔ (3) The norm satisfies the triangle (or Minkowski) inequality:

f + g 2 f 2 + g 2 . k kL ≤ k kL k kL (4) The linear space L2(Rn) is complete with respect to the norm

1 2 2 f L2 =( f(x) dx ) . k k ZRn | | | 2 Rn That is, if fn n∞=1 L ( ) is a Cauchy sequence of functions, then there { } ⊆ 2 n exists a limit f(x) L (R ) such that f (x) 2 f(x) (meaning ∈ n →L f (x) f(x) 2 0). k n − kL → (5) Furthermore, the norm on L2(Rn) is given by an inner :

2 (5.1) f,g = f(x)g(x) dx , f,f = f 2 . h i Z h i k kL The inner product satisfies the Schwartz inequality:

f,g f 2 g 2 . |h i| ≤ k kL k kL (6) The “”, or space of bounded linear functionals on L2(Rn), is L2(Rn) itself. Definition 5.3 (Banach and Hilbert spaces). A linear space with a norm, which is complete, is a . A Banach space whose norm is given by an inner product is a Hilbert space.

Our favourite Hilbert space L2(Rn) has its inner product defined in (5.1). One reason that L2(Rn) is a natural setting for the Fourier transform is that it is preserved under the transform. In Rn, we define the transform as follows: ˆ 1 iξ x f(ξ)= n e− · f(x)dx =( f)(ξ) . √2π ZRn F Theorem 5.4. The Fourier transform is an isometry of L2(Rn). That is, ˆ f L2(Rn) = f L2(Rn), or stated in other words, k k x k k ξ f(x) 2 dx = fˆ(ξ) 2 dξ . ZRn | | ZRn | | x ξ 5.1. Hilbert spaces 57

This equality between the L2 norms of a function and its Fourier transform is known as the Plancherel identity; it is a general fact about the Fourier transform that holds in many settings. The proof of Theorem 5.4 is deferred until the end of our discussion of Schwartz class. Other examples of Hilbert spaces and Banach spaces as tools of analysis include the following: (1) Sobolev spaces Hs(Rn) (Hilbert spaces based on L2 norms): 1 2 Hs(Rn)= f(x) : ∂αf(x) 2 dx < + . { Z | x | ∞} 0 Xα s   ≤| |≤ This is a scale of spaces, which are nested in terms of decreasing index s: Hs Hs 1 L2(Rn). ⊆ − ⊆···⊆ (2) Lp spaces (Banach spaces): 1 Lp(Rn)= f(x) : f(x) p dx p < + . { ZRn | |  ∞} For p = the space of bounded measurable functions on Rn is denoted by n ∞ L∞(R ), and has a norm given by

f L∞ := sup f(x) . k k x Rn | | ∈ (3) Sobolev spaces W s,p(Rn) (Banach spaces modeled on Lp norms): 1 W s,p(Rn)= f(x) : ∂αf(x) p dx p < + . { Z | x | ∞} 0 Xα s   ≤| |≤ (4) Schauder spaces Cs,γ(Ω) (Banach spaces modeled on C0 norms): s,γ α C (Ω) = f(x) : sup ∂x f(x) < + α s { x Ω | | ∞ ∀| |≤ ∈ α α ∂x f(x) ∂x f(y) and α = s, sup | − γ | < + . ∀| | x,y Ω,x=y x y } ∞} ∈ 6 | − | The many relationships of inclusion among these spaces, and their as- sociated inequalities of norms, are important information for the analysis of PDE. A principal one that is very often used in PDEs is the Sobolev Embedding Theorem. A version of this result is as follows: Theorem 5.5 (Sobolev embedding). If f(x) Hs(Rn) for a Sobolev index n ∈ s> 2 then f(x) ∞ C f s . | |L ≤ nk kH In other words Hs L for s > n , and there is a bound on the inclusion ⊆ ∞ 2 ι : Hs L given by the C . → ∞ n 58 5. Properties of the Fourier transform

Proof. Consider the value of f(x) at an arbitrary point of Rn,

1 iξ x ˆ f(x) = n e · f(ξ) dξ | | √ ZRn 2π 1 1 s iξ x 2 2 ˆ = n e · s (1 + ξ ) f(ξ) dξ . √ ZRn 2 2 | | 2π (1 + ξ ) | | Using the Cauchy – Schwarz inequality on this last expression 1 1 1 1 2 2 s ˆ 2 2 (5.2) f(x) n dξ (1 + ξ ) f(ξ) dξ . | |≤ √2π Z (1 + ξ 2)s Z | | | |  | |    By the binomial theorem 1 s 1 2 s 2 (1 + ξ 2)s fˆ(ξ) 2 dξ = ξ 2j fˆ(ξ) 2 dξ Z | | | | Z j| | | |   Xj=0  s 1 s j 2 2 = f(x) dx C f Hs , Z j|∇ | ≤ k k Xj=0  where we have used the Plancherel identity for f(x) and its in the last line. This holds of course for any s. It remins to bound the RHS of (5.2); 1 + 1 2 ∞ 1 n 1 2 s dξ = 2 s r − dr dSϕ , Z (1 + ξ ) ZSn−1 Z (1 + r )  | |  0 n  and if s> 2 then this quantity is finite, which finishes the proof.

5.2. Schwartz class

This linear space of functions was introduced by (Ecole Polytechnique - Paris) in order to embody a very convenient class of func- tions with which to work; for which one can interchange integrations with differentiations and integrate by parts with impunity. Colloquially, f(x) is a Schwartz class function if it is infinitely differentiable, and if it decays along with all of its derivatives as x faster than any . | | → ∞ Definition 5.6 (Schwartz class). Consider the functions f(x) defined over Rn such that for all α,β, α β sup x ∂x f(x) < + . x Rn | | ∞ ∈ The linear space of such functions is called Schwartz class.

Recall our previously introduced notation that is a convenient notational device for multivariable processes:; the quantities α,β are multi-indices, α = (α ,α ,...,α ), β = (β ,β ,...,β ), with each α ,β N. Thus, we have 1 2 n 1 2 n i j ∈ 5.2. Schwartz class 59

α α1 α2 αn α = α1 + + αn so that x = x1 x2 ...xn a monomial of degree α | | β| | β···1 β2 | |βn βj | | and ∂x = ∂x1 ∂x2 ...∂x = Π∂x a differential operator of order β . n j | | With this notation, many combinatorially complex objects can be writ- ten very conveniently. For example, Taylor series in n dimensions can be stated 1 f(x) ∂αf(y)(x y)α ∼ α! x − Xα with α!= α1!α2! ...αn!. Here are several examples of particular functions, where we can compare their properties of smoothness and decay in x Rn to those of Schwartz ∈ class: 2 x f(x)= e− 2 , ∈S 2 b 2 n f(x)=(1+ x )− 2 / but f(x) L (R ) for b > n/2 , | | ∈S ∈ x f(x)= e−| | / for a different reason . ∈S The space C0∞ of infinitely differentiable functions with compact is a subset of as a subspace, C , and the space C of bounded S 0∞ ⊆ S ∞ infinitely differentiable functions contains Schwartz class; C . The S ⊆ ∞ space Cω(Rn) of bounded real on Rn (the restriction to x Rn of functions that are analytic in a neighborhood of Rn Cn) is a ∈ ⊆ subset of C∞, but not every C∞ function is analytic. The topology of is defined using a countable family of seminorms S α β f α,β = sup x ∂x f(x) . k k x Rn | | ∈ That is to say, a sequence f (x) converges to f(x) if and only if for all { n }n∞=1 multi-indices α,β the seminorms α β fn f α,β = sup x ∂x (fn(x) f(x)) 0 k − k x | − | → with n . No uniformity in α,β is implied, however. Some functional → ∞ analytic remarks are in order at this point. The space is a linear space, but S not a Banach space, and there is no way to describe this sense of convergence by using a norm. It is, however, a metric space, with the distance between two points given by 1 f g ρ(f,g)= α,β , n( α + β ) k − k 2 1+ f g α,β Xα,β | | | | k − k and it is complete with respect to this metric. Clearly f (x) converges { n }n∞=1 in to f(x) if and only if ρ(f ,f) 0 if and only if all f f 0 as S n → k n − kα,β → n . The notation for limits of sequences of functions in this Schwartz → ∞ class sense is to write lim f = f. Metric spaces such as this one whose S− n 60 5. Properties of the Fourier transform topology is given by a countable number of semi-norms are called Fr´echet spaces. Lemma 5.7 (technical lemma). This result quantifies the propertes of scal- ing limits of two quantities, giving rise to well defined functionals on : S (i) Let g with g(0) = 1. Then for all f , ∈S ∈S lim (g(εx)f(x)) = f(x) . S− ε 0 → (ii) Let h(x) L1(Rn), with h(x)dx = 1, and suppose that f(x) is ∈ bounded, and continuous at x =R 0. Then 1 x lim f(x) h( )dx = f(0) . ε 0 ε ε → Z For h(x) L1Rn, with h(x) dx = 1, the limit in (5.7) describes ∈ the approximation of theR Dirac δ-function by rescaling a L1 function h(x).

Proof of Lemma 5.7(ii). By change of variables, 1 x g( ) dx = g(x′) dx′ = 1 , Z εn ε Z with x = x . Given δ > 0, choose r > 0 such that f(x) f(0) < δ for all ′ ε | − | x



The space of distributions is the space of continuous linear functionals on C ; = (C ) , otherwise known as its dual space. The space of 0∞ D′ 0∞ ′ tempered distributions is the dual space of . Also, we write for the dual S E′ space to C∞, the space of bounded infinitely differentiable functions. Since C C , then . 0∞ ⊆S⊆ ∞ E⊆S′ ⊆D′ The Fourier transform acts nicely on ; for f(x) define the Fourier S ∈S transform as usual

ˆ 1 iξ y f(ξ)= n e− · f(y) dy = (f)(ξ) . √2π Z F One checks that this converges absolutely because f(x) ; indeed ∈S iξ x 1 e− · f(x) f N,0 N , | | ≤ k k (1 + x 2) 2 | | and N >n + 1 will do for this. For Schwartz class f(x) we may exchange integrations and differentiations as much as we wish, therefore the following list of properties holds for fˆ(ξ).

Proposition 5.9 (Elementary properties of the Fourier transform on ). S (1) ∂xf(ξ)=(iξ)fˆ(ξ).

(2) xfd(ξ)= i∂ξfˆ(ξ). Noticec the vectorial notation; x and i∂ξ are vector operations. Define two operations on functions: τ f(x) = f(x h) (translations) and h − (σλf)(x)= f(λx) (dilations).

ihξ (3) τhf(ξ)= e− fˆ(ξ) a position boost. [ (4) edihxf(ξ)= fˆ(ξ h) a boost. − 1 ˆ 1 ˆ (5) σλf(ξ)= λ n σ(1/λ)(f)(ξ)= λ n f(ξ/λ). | | | | d The principal reason why Schwartz class is well-suited for Fourier anal- ysis is that it is invariant under the Fourier transform.

Theorem 5.10. The Fourier transform maps onto itself; that is, when- S ever f , then fˆ = (f) . ∈S F ∈S Proof. For f(x) , at the very least fˆ(ξ) makes sense as a convergent ∈ S integral. In order to check that fˆ(ξ) , we have to check that all of ∈ S the seminorms are finite, which we do using the properties described in 62 5. Properties of the Fourier transform

Proposition 5.9. ˆ α β ˆ f α,β = sup ξ ∂ξ f(ξ) k k ξ | |

1 iξ x 1 α 1 β = sup n e− · (( ∂x) ( x) f(x)) dx . ξ √2π Z i i

2 N This integrand is still bounded by C (1+ x )− 2 , and hence the supre- β+N,α | | mum over ξ Rn is finite. Furthermore, the Fourier transform is continuous ∈ on . Suppose that f (x) , and f f. This implies that for each S { n }n∞=1 n → α,β, S

ˆ ˆ 1 iξ x 1 α 1 β fn f α,β = sup n e− · (( ∂x) ( x) (fn(x) f(x))) dx 0 k − k ξ √2π Z i i − →

with n as well. Hence fˆ (ξ) fˆ(ξ), and therefore the mapping → ∞ n → S fˆ : is continuous.  S→S Finally we are prepared to prove the central theorem of Fourier inversion on . S Theorem 5.11 (Fourier inversion theorem on ). Suppose that f , then S ∈S 1 iξ x ˆ 1 ˆ f(x)= n e · f(ξ) dξ = − (f)(x) . √2π Z F

Proof. Lets first note that this works if f(x) is a Gaussian. For G(x) = 2 2 |x| |ξ| 1 e 2 then Gˆ(ξ)= 1 e 2 , by explicit calculation. One interpretation √2π − √2π − of this fact is that G(x) is an of the Fourier transform, (G)= F G, with eigenvalue 1. Now for the case of general f(x) , use properties ∈ S given in (5.7), 1 x 1 f(0) = lim f(x) G( ) dx = lim f(x) σ x (G) dx ε 0 εn ε ε 0 εn ε → Z → Z 1 \ = lim f(x) σ x (Gˆ) dx = lim f(x)σε(G) dx ε 0 εn ε ε 0 → Z → Z

= lim fˆ(ξ)σε(G)(ξ) dξ . ε 0 → Z We have to verify the last step of this sequence of calculations.

Lemma 5.12. For f,g , then ∈S f(x)gˆ(x)dx = fˆ(ξ)g(ξ)dξ . Z Z 5.2. Schwartz class 63

Proof of Lemma 5.12.

1 +iξ x 1 +iξ x f(x) n e · dx g(ξ) dξ = n f(x) e · dx g(ξ) dξ Z √2π Z √2π Z Z = (fˆ(ξ)g(ξ)) dx . Z 

1 Now apply (5.7) to the result. Since G(ξ) = n ; ξ=0 √2π

ˆ 1 ˆ f(0) = lim f(ε)G(ε/ξ) dξ = n f(ξ) dξ . ε 0 √ → Z 2π Z 1 This recovers f(0) = n fˆ(ξ) dξ. To generalize this procedure to cover all √2π x Rn is in fact simple, againR using the list of properties in Proposition 5.9: ∈ 1 1 iξ x ˆ f(x)=(τ xf)(0) = n (τ xf)(ξ) dξ = n e · f(ξ) dξ . − √2π Z F − √2π Z 

We can finish this circle of ideas with a proof of the L2(Rn) Fourier inversion theorem. We have shown that Schwartz class L2(Rn) is a S ⊆ dense subspace. The Lemma 5.12 states that for f,g , then ∈S 1 f, − g = f,g , h F i hF i where we also note that 1 ( − g)(x) =g ˆ( ξ) . F − Therefore for every f , ∈S 1 f 2 = f, − ( f) k kL h F F i = (f), (f) hF F i 2 = fˆ 2 . k kL so the Fourier transform acts on the the subspace L2(Rn) isometrically S ⊆ (in the sense of the L2-norm). Since is dense, given an arbitrary f S ∈ L2(Rn), there is a sequence f f in the L2(Rn) sense of convergence, n → with f . Then we define n ∈S

(f)= lim (fn). n F →∞ F Theorem 5.13 (of ). : Suppose that an operator T is defined on a dense subspace a Banach space, and it is bounded; S ⊆B Tf C f for all f . Then there exists a unique extension T(1) k kB ≤ k kB ∈ S of T to all of , which is bounded with the same bound. B 64 5. Properties of the Fourier transform

Proof. For arbitrary f consider a sequence f f, with f ∈B { n}n∞=1 → { n}n∞=1 ⊆ . Then f is Cauchy, and so is the sequence Tf because S { n}n∞=1 { n}n∞=1 Tfn Tfm C fn fm 0 . k − kB ≤ k − kB → Thus Tf has a limit g , which we define to be T(1)f := g. The { n}n∞=1 ∈ B extension is clearly linear. It is also well-defined, for if h is { n}n∞=1 ⊆ S another sequence such that h f, but Th g , then { n}n∞=1 → n → 1 g g1 = lim Tfn Thn C lim fn hn = 0 . B n B n B k − k →∞ k − k ≤ →∞ k − k 

The conclusion that is relevant to the Fourier transform is that re- F stricted to (such that it has norm f 2 = f 2 on ) extends uniquely S kF kL k kL S to (1) defined on all of L2(Rn). Furthermore, this extension is an isometry F of L2(Rn), as for f L2(Rn), with f f; { n}n∞=1 ⊆S⊆ n → 2 2 2 1 2 f 2 = lim fn 2 = lim fˆn 2 = (f) 2 . L n L n L L k k →∞ k k →∞ k k kF k This completes the proof of 5.4. We will redefine the extension notation (1) := at this time, dropping the extra superscript. The statement is F F 2 then that f 2 = f 2 for all f L , which is the desired result. k kL kF kL ∈ To finish our remarks we should note that complex isometries U have 1 the property that U∗ = U− ; namely they are unitary operators. Lets verify this with the Fourier transform.

Definition 5.14. The adjoint T∗ of an operator T on a Hilbert space H satisfies T f,g = f, Tg for all f,g H. h ∗ i h i ∈ Applying this definition of the adjoint to , take any two f,g L2. F ∈ Then

1 iξ x g(ξ) (f)(ξ) dξ = g(ξ) n e− · f(x) dxdξ ZRn F ZRn √2π Z 1 iξ x = ( n e · g(ξ) dξ)f(x) dx ZRn ZRn √2π 1 = − (g)(x)(f(x)) dx , ZRn F which states that = 1, as desired. F ∗ F − Exercises: Chapter 5

Exercise 5.1. Show that the Fourier transform preserves angles between vectors. For f,g L2 then, ∈ Re f,g = f 2 g 2 cos θ h i k kL k kL Exercises: Chapter 5 65 and Re f,ˆ gˆ = fˆ 2 gˆ 2 cos ϕ . h i k kL k kL Show that θ = ϕ. Exercise 5.2. The Lp(Ω) norm for functions on a domain Ω Rn is defined ⊆ as 1/p p f Lp = f(x) dx , k k ZΩ | |  for 1 p< + . The H¨older inequality states that ≤ ∞

f(x)g(x) dx f Lp g p′ Z ≤ k k k kL Ω 1 1 where p + p′ = 1 are dual indices. A special case for p = p′ = 2 is the Cauchy – Schwarz inequality.

Give a proof of the H¨older inequality for the range of p,p′ given above. Exercise 5.3. This problem concerns the case of domains Ω = Rn and of bounded domains Ω Rn. ⊆ (1) In the case of Ω = Rn, for which indices p and q does the following hold: Lp(Rn) Lq(Rn) ? ⊆ (2) In the case of bounded Ω, use the H¨older inequality to show that Lp(Ω) Lq(Ω) . ⊆ for q p. ≤ (3) In the case of bounded Ω, is it true that p L∞(Ω) = L (Ω) . 1 p<\+ ≤ ∞ Exercise 5.4. Prove the part (i) of the technical lemma 5.7 on Schwartz class functins. Namely show that for g (Rn) such that g(0) = 1, then ∈ S for all f (Rn) ∈S lim g(εx)f(x) = f(x) . S− ε 0 →  n Exercise 5.5. Give examples of non-analytic, C∞ functions on R . Exercise 5.6. We have described the Fourier transform as a unitary op- 2 n F erator on L (R ). What are the eigenvalues and (λk,ψk(x)) of this operator. For simplicity you may consider only the case n = 1. Exercise 5.7. The Schr¨odinger operator that describes quantum harmonic oscillator is given by 1 ∆ψ 1 x 2ψ = λψ . 2 − 2 | | Show that it is invariant under the Fourier transform, and describe its eigen- functions and eigenvalues.