Fourier Transform

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15 Fourier Transform

In this section we will briefly present a theory of Fourier transform focusing on commutative group approach. We mainly follow footsteps of []*Ch. IV.

15.1 Convolutions on Commutative Groups

Let G be a commutative group, we will use + sign to denote group operation, respectively the inverse elements of g∈ G will be denoted −g. We assume that G has a Hausdorff topology such that operations (g₁,g₂)↦ g₁+g₂ and g↦ −g are continuous maps. We also assume that the topology is locally compact, that is the group neutral element has a neighbourhood with a compact closure.

Example 1 Our main examples will be as follows:

G=ℤ the group of integers with operation of addition and the discrete topology (each point is an open set).
G=ℝ the group of real numbers with addition and the topology defined by open intervals.
G=T the group of Euclidean rotations the unit circle in ℝ² with the natural topology. Another realisations of the same group:
- Unimodular complex numbers under multiplication.
- Factor group ℝ/ℤ, that is addition of real numbers modulo 1.
There is a homomorphism between two realisations given by z=e^{2πi t}, t∈[0,1), | z |=1.

We assume that G has a regular Borel measure which is invariant in the following sense.

Definition 2 Let µ be a measure on a commutative group G, µ is called invariant (or Haar measure) if for any measurable X and any g∈ G the sets g+X and −X are also measurable and µ(X)=µ(g+X)=µ(−X).

Such an invariant measure exists if and only if the group is locally compact, in this case the measure is uniquely defined up to the constant factor.

Exercise 3 Check that in the above three cases invariant measures are:

G=ℤ, the invariant measure of X is equal to number of elements in X.
G=ℝ the invariant measure is the Lebesgue measure.
G=T the invariant measure coincides with the Lebesgue measure.

Definition 4 A convolution of two functions on a commutative group G with an invariant measure µ is defined by:

(f₁*f₂)(x)=

∫

f₁(x−y) f₂(y) d µ(y)=

∫

f₁(y) f₂(x−y) d µ(y). (83)

Theorem 5 If f₁, f₂∈L₁(G,µ), then the integrals in (83) exist for almost every x∈ G, the function f₁*f₂ is in L₁(G,µ) and ||f₁*f₂||≤ ||f₁||·||f₂||.

Proof. If f₁, f₂∈L₁(G,µ) then by Fubini’s Thm. 50 the function φ(x,y)=f₁(x)*f₂(y) is in L₁(G× G, µ× µ) and ||φ||=||f₁||· ||f₂||.

Let us define a map τ: G× G → G× G such that τ(x,y)=(x+y,y). It is measurable (send Borel sets to Borel sets) and preserves the measure µ×µ. Indeed, for an elementary set C=A× B⊂ G× G we have:

(µ×µ)(τ(C))

∫

G× G

χ_τ(C)(x,y) d µ(x) d µ(y)

∫

G× G

χ_C(x−y,y) d µ(x) d µ(y)

∫

⎛
⎜
⎜
⎜
⎜
⎝

∫

χ_C(x−y,y) d µ(x)

⎞
⎟
⎟
⎟
⎟
⎠

dµ(y)

∫

µ(A+y) d µ(y)=µ(A)× µ(B)=(µ×µ)(C).

We used invariance of µ and Fubini’s Thm. 50. Therefore we have an isometric isomorphism of L₁(G× G,µ× µ) into itself by the formula:

Tφ(x,y)=φ(τ(x,y))=φ(x−y,y).

If we apply this isomorphism to the above function φ(x,y)=f₁(x)*f₂(y) we shall obtain the statement. □

Definition 6 Denote by S(k) the map S(k): f↦ k*f which we will call convolution operator with the kernel k.

Corollary 7 If k∈L₁(G) then the convolution S(k) is a bounded linear operator on L₁(G).

Theorem 8 Convolution is a commutative, associative and distributive operation. In particular S(f₁)S(f₂)=S(f₂)S(f₁)=S(f₁*f₂).

Proof. Direct calculation using change of variables. □

It follows from Thm. 5 that convolution is a closed operation on L₁(G) and has nice properties due to Thm. 8. We fix this in the following definition.

Definition 9 L₁(G) equipped with the operation of convolution is called convolution algebra L₁(G).

The following operators of special interest.

Definition 10 An operator of shift T(a) acts on functions by T(a): f(x)↦ f(x+a).

Lemma 11 An operator of shift is an isometry of L_p(G), 1≤ p≤∞.

Theorem 12 Operators of shifts and convolutions commute:

T(a)(f₁*f₂)=T(a)f₁*f₂=f₁*T(a)f₂,

T(a)S(f)=S(f)T(a)=S(T(a)f).

Proof. Just another calculation with a change of variables. □

Remark 13 Note that operator of shifts T(a) provide a representation of the group G by linear isometric operators in L_p(G), 1≤ p≤ ∞. A map f↦ S(f) is a representation of the convolution algebra

There is a useful relation between support of functions and their convolutions.

Lemma 14 For any f₁, f₂∈L₁(G) we have:

supp(f₁*f₂)⊂supp(f₁)+supp(f₂).

Proof. If x∉supp(f₁)+supp(f₂) then for any y∈supp(f₂) we have x−y∉supp(f₁). Thus for such x convolution is the integral of the identical zero. □

Exercise 15 Suppose that the function f₁ is compactly supported and k times continuously differentiate in ℝ, and that the function f₂ belongs to L₁(ℝ). Prove that the convolution f₁* f₂ has continuous derivatives up to order k.
[Hint: Express the derivative d/d x as the limit of operators (T(h)−I)/h when h→ 0 and use Thm. 12.]

15.2 Characters of Commutative Groups

Our purpose is to map the commutative algebra of convolutions to a commutative algebra of functions with point-wise multiplication. To this end we first represent elements of the group as operators of multiplication.

Definition 16 A character χ: G→ T is a continuous homomorphism of an abelian topological group G to the group T of unimodular complex numbers under multiplications:

χ(x+y)=χ(x)χ(y).

Note, that a character is an eigenfunction for a shift operator T(a) with the eigenvalue χ(a). Furthermore, if a function f on G is an eigenfunction for all shift operators T(a), a∈ G then the collection of respective eigenvalues λ(a) is a homomorphism of G to ℂ and f(a)=α λ(a) for some α∈ℂ. Moreover, if T(a) act by isometries on the space containing f(a) then λ(a) is a homomorphism to T.

Lemma 17 The product of two characters of a group is again a character of the group. If χ is a character of G then χ⁻¹=χ is a character as well.

Proof. Let χ₁ and χ₂ be characters of G. Then:

χ₁(gh)χ₂(gh)	=	χ₁(g)χ₁(h)χ₂(g)χ₂(h)
	=	(χ₁(g)χ₂(g))(χ₁(h)χ₂(h))∈T.

□

Definition 18 The dual group Ĝ is collection of all characters of G with operation of multiplication.

The dual group becomes a topological group with the uniform convergence on compacts: for any compact subset K⊂ G and any ε>0 there is N∈ℕ such that | χ_n(x)−χ(x) |<ε for all x∈ K and n>N.

Exercise 19 Check that

The sequence f_n(x)=xⁿ does not converge uniformly on compacts if considered on [0,1]. However it does converges uniformly on compacts if considered on (0,1).
If X is a compact set then the topology of uniform convergence on compacts and the topology uniform convergence on X coincide.

Example 20 If G=ℤ then any character χ is defined by its values χ(1) since

χ(n)=[χ(1)]ⁿ. (84)

Since χ(1) can be any number on T we see that $ℤ^_$ is parametrised by T.

Theorem 21 The group $ℤ^_$ is isomorphic to T.

Proof. The correspondence from the above example is a group homomorphism. Indeed if χ_z is the character with χ_z(1)=z, then χ_z₁χ_z₂=χ_{z₁ z₂}. Since ℤ is discrete, every compact consists of a finite number of points, thus uniform convergence on compacts means point-wise convergence. The equation (84) shows that χ_{z_n}→ χ_z if and only if χ_{z_n}(1)→ χ_z(1), that is z_n→ z. □

Theorem 22 The group $T^_$ is isomorphic to ℤ.

Proof. For every n∈ℤ define a character of T by the identity

χ_n(z)=zⁿ, z∈T. (85)

We will show that these are the only characters in Cor. 26. The isomorphism property is easy to establish. The topological isomorphism follows from discreteness of $T^_$. Indeed due to compactness of T for n≠ m:

max

z∈T

⎪
⎪

χ_n(z)−χ_m(z)

⎪
⎪

²=

max

z∈T

⎪
⎪

1−ℜ z^m−n

⎪
⎪

²=2²=4.

Thus, any convergent sequence (n_k) have to be constant for sufficiently large k, that corresponds to a discrete topology on ℤ. □

The two last Theorem are an illustration to the following general statement.

Principle 23 (Pontryagin’s duality) For any locally compact commutative topological group G the natural map G→ Ĝ, such that it maps g∈ G to a character f_g on Ĝ by the formula:

f_g(χ)=χ(g), χ∈Ĝ, (86)

is an isomorphism of topological groups.

Remark 24

The principle is not true for commutative group which are not locally compact.
Note the similarity with an embedding of a vector space into the second dual.

In particular, the Pontryagin’s duality tells that the collection of all characters contains enough information to rebuild the initial group.

Theorem 25 The group $ℝ^_$ is isomorphic to ℝ.

Proof. For λ∈ℝ define a character χ_λ∈$ℝ^_$ by the identity

χ_λ(x)=e^{2π i λ x}, x∈ℝ. (87)

Moreover any smooth character of the group G=(ℝ, +) has the form (87). Indeed, let χ be a smooth character of ℝ. Put c=χ′(t)|_t=0∈ ℂ. Then χ′(t)=cχ(t) and χ(t)=e^ct. We also get c∈ iℝ and any such c defines a character. Then the multiplication of characters is: χ₁(t)χ₂(t)=e^c₁te^c₂t=e^(c₂+c₁)t. So we have a group isomorphism.

For a generic character we can apply first the smoothing technique and reduce to the above case.

Let us show topological homeomorphism. If λ_n→ λ then χ_{λ_n}→ χ_λ uniformly on any compact in ℝ from the explicit formula of the character. Reverse, let χ_{λ_n}→ χ_λ uniformly on any interval. Then χ_{λ_n−λ}(x)→ 1 uniformly on any compact, in particular, on [0,1]. But

sup

[0,1]

⎪
⎪

χ_{λ_n−λ}(x)− 1

⎪
⎪

sup

[0,1]

⎪
⎪

sinπ (λ_n−λ)x

⎪
⎪

⎧
⎪
⎨
⎪
⎩

⎪
⎪

λ_n−λ

⎪
⎪

≥ 1/2,

sinπ

⎪
⎪

λ_n−λ

⎪
⎪

λ_n−λ

⎪
⎪

≤ 1/2.

Thus λ_n→ λ. □

Corollary 26 Any character of the group T has the form (85).

Proof. Let χ∈$T^_$, consider χ₁(t)=χ(e^{2π
i t}) which is a character of ℝ. Thus χ₁(t)=e^{2π i λ t} for some λ∈ℝ. Since χ₁(1)=1 then λ=n∈ℤ. Thus χ₁(t)=e^{2π i n
t}, that is χ(z)=zⁿ for z=e^{2π i t}. □

Remark 27 Although $ℝ^_$ is isomorphic to ℝ there is no a canonical form for this isomorphism (unlike for ℝ→ $ℝ^_$). Our choice is convenient for the Poisson formula below, however some other popular definitions are λ→ e^{i λ x} or λ→ e^{−i λ x}.

We can unify the previous three Theorem into the following statement.

Theorem 28 Let G=ℝⁿ× ℤ^k× T^l be the direct product of groups. Then the dual group is Ĝ=ℝⁿ× T^k× ℤ^l.

15.3 Fourier Transform on Commutative Groups

Definition 29 Let G be a locally compact commutative group with an invariant measure µ. For any f∈L₁(G) define the Fourier transform f by

f(χ)=

∫

f(x) χ(x) d µ(x), χ∈Ĝ. (88)

That is the Fourier transform f is a function on the dual group Ĝ.

Example 30

If G=ℤ, then f∈L₁(Z) is a two-sided summable sequence (c_n)_n∈ℤ. Its Fourier transform is the function f(z)=∑_n=−∞^∞c_n zⁿ on T. Sometimes f(z) is called generating function of the sequence (c_n).
If G=T, then the Fourier transform of f∈L₁(T) is its Fourier coefficients, see Section 5.1.
If G=ℝ, the Fourier transform is also the function on ℝ given by the Fourier integral:
f(λ)= ∫

ℝ

f(x)  e^{−2πiλ x} d x. (89)

The important properties of the Fourier transform are captured in the following statement.

Theorem 31 Let G be a locally compact commutative group with an invariant measure µ. The Fourier transform maps functions from L₁(G) to continuous bounded functions on Ĝ. Moreover, a convolution is transformed to point-wise multiplication:

(f₁*f₂)^ (χ)=f₁(χ)·f₂(χ), (90)

a shift operator T(a), a∈ G is transformed in multiplication by the character f_a∈Ĝ:

(T(a)f)^ (χ)=f_a(χ)·f(χ), f_a(χ)=χ(a) (91)

and multiplication by a character χ∈Ĝ is transformed to the shift T(χ⁻¹):

(χ· f)^ (χ₁)=T(χ⁻¹)f(χ₁)=f(χ⁻¹χ₁). (92)

Proof. Let f∈L₁(G). For any ε>0 there is a compact K⊂ G such that ∫_{G∖ K} | f | d µ<ε. If χ_n→ χ in Ĝ, then we have the uniform convergence of χ_n→ χ on K, so there is n(ε) such that for k>n(ε) we have | χ_k(x)−χ(x) | <ε for all x∈ K. Then

⎪
⎪

f(χ_n)−f(χ)

⎪
⎪

≤

∫

⎪
⎪

f(x)

⎪
⎪

χ_n(x)− χ(x)

⎪
⎪

 d µ(x)+

∫

G∖ K

⎪
⎪

f(x)

⎪
⎪

χ_n(x)− χ(x)

⎪
⎪

 d µ(x)

≤

⎪⎪
⎪⎪

+2ε.

Thus f is continuous. Its boundedness follows from the integral estimations. Algebraic maps (90)–(92) can be obtained by changes of variables under integration. For example, using Fubini’s Thm. 50 and invariance of the measure:

(f₁*f₂)^ (χ )

∫

f₁(s) f₂(t−s) d s χ(t) d t

∫

f₁(s) χ(s) f₂(t−s) χ(t−s) d s d t

f₁(χ)f₂(χ).

□

15.4 The Schwartz space of smooth rapidly decreasing functions

We say that a function f is rapidly decreasing if lim_{x→ ±∞} | x^kf(x) |=0 for any k∈ℕ.

Definition 32 The Schwartz space denoted by S or space of rapidly decreasing functions on Rn is the space of infinitely differentiable functions such that:

S =

⎧
⎪
⎨
⎪
⎩

f∈ C^∞(ℝ):

sup

x∈ ℝ

⎪
⎪

x^αf^(β)(x)

⎪
⎪

<∞ ∀ α ,β ∈ ℕ

⎫
⎪
⎬
⎪
⎭

. (93)

Example 33 An example of a rapidly decreasing function is the Gaussian e^{−π x²}.

It is worth to notice that S⊂ L_p(ℝ) for any 1<p<∞. Moreover, S is dense in L_p(ℝ), for p=1 this can be shown in the following steps (other values of p can be done similarly but require some more care). First we will show that S is an ideal of the convolution algebra L₁(ℝ).

Exercise 34 For any g∈ S and f ∈ L₁(ℝ) with compact support their convolution f*g belongs to S. [Hint: smoothness follows from Ex. 15.]

Define the family of functions g_t(x) for t>0 in S by scaling the Gaussian:

g_t(x) =

e^{−π (x/t)²}.

Exercise 35 Show that g_t(x) satisfies the following properties, cf. Lem 7:

g_t(x)>0 for all x∈ℝ and t>0.
∫_ℝ g_t(x) d x=1 for all t>0. [Hint: use the table integral ∫_ℝ e^−π
x² d x=1.]
For any ε>0 and any δ>0 there exists T>0 such that for all positive t< T we have:
0<
−δ

∫

−∞

+
∞

∫

δ

g_t(x) d x < ε.

It is easy to see, that the above properties 1–3 are not unique to the Gaussian and a wide class have them. Such a family a family of functions is known as approximation of the identity [] due to the next property (94).

Exercise 36

Let f be a continuous function with compact support, then

lim

t→ 0

⎪⎪
⎪⎪ f−g_t*f ⎪⎪
⎪⎪ ₁=0 . (94)

[Hint: use the proof of Thm. 8.]
The Schwartz space S is dense in L₁(ℝ). [Hint: use Prop. 20, Ex. 34 and (94).]

15.5 Fourier Integral

We recall the formula (89):

Definition 37 We define the Fourier integral of a function f∈ L₁(ℝ) by

f(λ)=

∫

ℝ

f(x) e^{−2π i λ
x} d x. (95)

We already know that f is a bounded continuous function on ℝ, a further property is:

Lemma 38 If a sequence of functions (f_n)⊂L₁(ℝ) converges in the metric L₁(ℝ), then the sequence (f_n) converges uniformly on the real line.

Proof. This follows from the estimation:

⎪
⎪

f_n(λ)−f_m(λ)

⎪
⎪

≤

∫

ℝ

⎪
⎪

f_n(x)−f_m(x)

⎪
⎪

 d x.

□

Lemma 39 The Fourier integral f of f∈L₁(ℝ) has zero limits at −∞ and +∞.

Proof. Take f the indicator function of [a,b]. Then f(λ )=1/−2π i λ (e^{−2π i a} −e^{−2π i
b}), λ≠ 0. Thus lim_λ→
±∞ f(λ)=0. By continuity from the previous Lemma this can be extended to the closure of step functions, which is the space L₁(ℝ) by Lem. 17. □

Lemma 40 If f is absolutely continuous on every interval and f′∈L₁(ℝ), then

(f′)^=2πi λ f.

More generally:

(f^(k))^=(2πi λ)^kf. (96)

Proof. A direct demonstration is based on integration by parts, which is possible because assumption in the Lemma.

It may be also interesting to mention that the operation of differentiation D can be expressed through the shift operatot T_a:

lim

Δ t → 0

T_Δ t− I

Δ t

. (97)

By the formula (91), the Fourier integral transforms 1/Δ t(T_{Δ t}− I) into 1/Δ t(χ_λ(Δ t)− 1). Providing we can justify that the Fourier integral commutes with the limit, the last operation is multiplication by χ′_λ(0)=2πi λ. □

Corollary 41 If f^(k)∈L₁(ℝ) then

⎪
⎪

(f^(k))^

⎪
⎪

2π λ

⎪
⎪

→ 0 as λ → ∞,

that is f decrease at infinity faster than | λ |^−k.

Lemma 42 Let f(x) and xf(x) are both in L₁(ℝ), then f is differentiable and

f′=(−2 π i xf)^.

More generally

f^(k)=((−2π i x)^k f)^. (98)

Proof. There are several strategies to prove this results, all having their own merits:

The most straightforward uses the differentiation under the integration sign.
We can use the intertwining property (92) of the Fourier integral and the connection of derivative with shifts (97).
Using the inverse Fourier integral (see below), we regard this Lemma as the dual to the Lemma 40.

□

Corollary 43 The Fourier transform of a smooth rapidly decreasing function is a smooth rapidly decreasing function.

Corollary 44 The Fourier integral of the Gaussian e^{−π x²} is e^{−π λ ²}.

Proof.[] Note that the Gaussian g(x)=e^{−π x²} is a unique (up to a factor) solution of the equation g′+2π x g=0. Then, by Lemmas 40 and 42, its Fourier transform shall satisfy to the equation 2πi λ ĝ+ iĝ′=0. Thus, ĝ=c· e^{−π λ
²} with a constant factor c, its value 1 can be found from the classical integral ∫_ℝ e^−π
x² d x=1 which represents ĝ(0). □

The relation (96) and (98) allows to reduce many partial differential equations to algebraic one, see § 0.2 and 5.4. To convert solutions of algebraic equations into required differential equations we need the inverse of the Fourier transform.

Definition 45 We define the inverse Fourier transform on L₁(ℝ):

f(λ)=

∫

ℝ

f(x) e^{2π i λ x} d x. (99)

We can notice the formal correspondence f(λ)=f(−λ)=f(λ), which is a manifestation of the group duality $ℝ^_$=ℝ for the real line. This immediately generates analogous results from Lem. 38 to Cor. 44 for the inverse Fourier transform.

Theorem 46 The Fourier integral and the inverse Fourier transform are inverse maps. That is, if g=f then f=ǧ.

Proof.[Sketch of a proof] The exact meaning of the statement depends from the spaces which we consider as the domain and the range. Various variants and their proofs can be found in the literature. For example, in [, § IV.2.3], it is proven for the Schwartz space S of smooth rapidly decreasing functions.

The outline of the proof is as follows. Using the intertwining relations (96) and (98), we conclude the composition of Fourier integral and the inverse Fourier transform commutes both with operator of multiplication by x and differentiation. Then we need a result, that any operator commuting with multiplication by x is an operator of multiplication by a function f. For this function, the commutation with differentiation implies f′=0, that is f=const. The value of this constant can be evaluated by a Fourier transform on a single function, say the Gaussian e^−π
x² from Cor. 44. □

The above Theorem states that the Fourier integral is an invertible map. For the Hilbert space L₂(ℝ) we can show a stronger property—its unitarity.

Theorem 47 (Plancherel identity) The Fourier transform extends uniquely to a unitary map L₂(ℝ)→ L₂(ℝ):

∫

ℝ

⎪
⎪

² d x=

∫

ℝ

⎪
⎪

² d λ. (100)

Proof. The proof will be done in three steps: first we establish the identity for smooth rapidly decreasing functions, then for L₂ functions with compact support and finally for any L₂ function.

Take f₁ and f₂∈S be smooth rapidly decreasing functions and g₁ and g₂ be their Fourier transform. Then (using Fubini’s Thm. 50):

∫

ℝ

f₁(t) f₂(t) d t

∫

ℝ

∫

ℝ

g₁(λ )  e^{2π i
λ t} d λ  f₂(t) d t

∫

ℝ

g₁(λ )

∫

ℝ

  e^{2π i
λ t}  f₂(t) d t d λ

∫

ℝ

g₁(λ )  ḡ₂(λ ) d λ

Put f₁=f₂=f (and therefore g₁=g₂=f) we get the identity ∫| f |² d x=∫| f |² d λ.

The same identity (100) can be obtained from the property (f₁f₂)^=f₁*f₂, cf. (90), or explicitly:

∫

ℝ

f₁(x) f₂(x) e^{−2π iλ x} d x=

∫

ℝ

f₁(t) f₂(λ−t) d t.

Now, substitute λ=0 and f₂=f₁ (with its corollary f₂(t)=f₁(−t)) and obtain (100).

Next let f∈L₂(ℝ) with a support in (−a,a) then f∈L₁(ℝ) as well, thus the Fourier transform is well-defined. Let f_n∈S be a sequence with support on (−a,a) which converges to f in L₂ and thus in L₁. The Fourier transform g_n converges to g uniformly and is a Cauchy sequence in L₂ due to the above identity. Thus g_n→ g in L₂ and we can extend the Plancherel identity by continuity to L₂ functions with compact support.
The final bit is done for a general f∈L₂ the sequence
f_n(x)= ⎧
⎪
⎨
⎪
⎩
f(x),
if ⎪
⎪ x ⎪
⎪ <n,

0, otherwise;

of truncations to the interval (−n,n). For f_n the Plancherel identity is established above, and f_n→ f in L₂(ℝ). We also build their Fourier images g_n and see that this is a Cauchy sequence in L₂(ℝ), so g_n→ g.

If f∈L₁∩L₂ then the above g coincides with the ordinary Fourier transform on L₁. □

We note that Plancherel identity and the Parseval’s identity (30) are cousins—they both states that the Fourier transform L₂(G)→ L₂(Ĝ) is an isometry for G=ℝ and G=T respectively. They may be combined to state the unitarity of the Fourier transform on L₂(G) for the group G=ℝⁿ× ℤ^k× T^l cf. Thm. 28.

Proofs of the following statements are not examinable Thms. 23, 36, 53, 33, 46, Props. 14, 20.


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