Obviously, the proof can be extended to the following situation.

Corollary 5.24 If f ∈ L 1 (Rd ) is continuous and has a Fourier transform f ∈ L 1 (Rd ) then

f can be recovered from its Fourier transform:

f (x) = (2π)’d/2 f (ω)ei x ω dω,

T

x ∈ Rd .

Rd

Another consequence of Theorem 5.23 is that it allows us to extend the idea of the Fourier

transform to L 2 (Rd ), even for functions that are not integrable and hence do not possess a

classical Fourier transform. Theorem 5.23 asserts that Fourier transformation constitutes a

bounded linear operator de¬ned on the dense subset S of L 2 (Rd ). Therefore, there exists

a unique bounded extension T of this operator to all L 2 (Rd ), which we will call Fourier

transformation on L 2 (Rd ). We will also use the notation f = T ( f ) for f ∈ L 2 (Rd ). In

general, f for f ∈ L 2 (Rd ) is given as the L 2 (Rd )-limit of { f n } if f n ∈ S converges to f in

L 2 (Rd ).

Corollary 5.25 (Plancherel) There exists an isomorphic mapping T : L 2 (Rd ) ’ L 2 (Rd )

such that:

(1) T f L 2 (R d ) = f L 2 (R d ) for all f ∈ L 2 (Rd );

(2) T f = f for all f ∈ L 2 (Rd ) © L 1 (Rd );

(3) T ’1 g = g ∨ for all g ∈ L 2 (Rd ) © L 1 (Rd ).

The isomorphism is uniquely determined by these properties.

60 Auxiliary tools from analysis and measure theory

Proof The proof follows from the explanation given in the paragraph above this corollary

and the fact that S ⊆ L 1 (Rd ) © L 2 (Rd ), so that L 1 (Rd ) © L 2 (Rd ) is also dense in L 2 (Rd )

with respect to the L 2 (Rd )-norm.

Finally, we will take a look at the Fourier transform of a radial function. Surprisingly, it

turns out to be radial as well. This is of enormous importance in the theory to come.

Theorem 5.26 Suppose ∈ L 1 (Rd ) © C(Rd ) is radial, i.e. (x) = φ( x 2 ), x ∈ Rd . Then

its Fourier transform is also radial, i.e. (ω) = Fd φ( ω 2 ) with

∞

Fd φ(r ) = r ’(d’2)/2 φ(t)t d/2 J(d’2)/2 (r t)dt.

0

Proof The case d = 1 follows immediately from

1/2

2

J’1/2 (t) = cos t.

πt

In the case d ≥ 2 we set r = x 2 . Splitting the Fourier integral and using (5.1) yields

(x) = (2π)’d/2 (ω)e’i x ω dω

T

Rd

∞

= (2π)’d/2 (t ω 2 )e’it x ω d S(ω)dt

T

t d’1

S d’1

0

∞

= (2π)’d/2 e’it x ω d S(ω)dt

T

φ(t)t d’1

S d’1

0

∞

= r ’(d’2)/2 φ(t)t d/2 J(d’2)/2 (r t)dt.

0

5.3 Measure theory

We assume the reader to have some familiarity with measure and integration theory. The

convergence results of Fatou, Beppo Levi, and Lebesgue for integrals de¬ned by general

measures should be known. Other results like Riesz™ representation theorem and Helly™s

theorem are perhaps not standard knowledge. Hence, we now review the material relevant

to us. The reader should be aware of the fact that terms like Borel measure and Radon

measure have different meanings throughout the literature. Hence, when using results from

measure theory it is crucial to ¬rst have a look at the de¬nitions. Here, we will mainly

use the de¬nitions and results of Bauer [9] since his de¬nition of a measure is to a certain

extent constructive. Another good source for the results here is Halmos [78]. We will not

give proofs in this short section. Instead, we refer the reader to the books just mentioned.

Moreover, Helly™s theorem can be found in the book [46] by Donoghue.

Let be an arbitrary set. We denote the set of all subsets of by P( ). We have to

introduce several names and concepts now.

5.3 Measure theory 61

De¬nition 5.27 A subset R of P( ) is called a ring on if

(1) … ∈ R,

(2) A, B ∈ R implies A \ B ∈ R,

(3) A, B ∈ R implies A ∪ B ∈ R.

The name is motivated by the fact that a ring R is indeed a ring in the algebraical sense

if one takes the intersection © as multiplication and the symmetric difference , de¬ned by

A B := (A \ B) ∪ (B \ A), as addition.

We are concerned with certain functions de¬ned on a ring.

De¬nition 5.28 Let R be a ring on a set . A function μ : R ’ [0, ∞] is called a pre-

measure if

(1) μ(…) = 0,

(2) for disjoint A j ∈ R, j ∈ N, with ∪A j ∈ R we have μ(∪A j ) = μ(A j ).

The last property is called σ -additivity.

Note that for the second property we consider only those sequences {A j } of disjoint sets

whose union is also contained in R. This property is not automatically satis¬ed for a ring.

It is different in the situation of a σ -algebra.

De¬nition 5.29 A subset A of P( ) is called a σ -algebra on if

∈ A,

(1)

(2) A ∈ A implies \ A ∈ A,

(3) A j ∈ A, j ∈ N, implies ∪ j∈N A j ∈ A.

Obviously, each σ -algebra is also a ring. Moreover, each ring R, or more generally each

subset R of P( ), de¬nes a smallest σ -algebra that contains R. This σ -algebra is denoted

by σ (R) and is obviously given by

σ (R) = ©{A : A is a σ -algebra and R ⊆ A}.

We also say that R generates the σ -algebra σ (R).

De¬nition 5.30 A pre-measure de¬ned on a σ -algebra is called a measure. The sets in the

σ -algebra are called measurable with respect to this measure.

Obviously measurability depends actually more on the σ -algebra than on the actual

measure.

Since any ring R is contained in σ (R) it is natural to ask whether a pre-measure μ on

R has an extension μ to σ (R), meaning that μ(A) = μ(A) for all A ∈ R. The answer is

af¬rmative.

Proposition 5.31 Each pre-measure μ on a ring R on has an extension μ to σ (R).

The measures introduced so far will also be called nonnegative measures in contrast with

signed measures. A signed measure is a function μ : A ’ R = R ∪ {’∞, ∞} de¬ned on

a σ -algebra A, which is σ -additive but not necessarily nonnegative.

62 Auxiliary tools from analysis and measure theory

A measure is called ¬nite if μ( ) < ∞. The total mass of a nonnegative measure is

given by μ := μ( ). A signed measure μ can be decomposed into μ = μ+ ’ μ’ with

two nonnegative measures μ+ , μ’ . In this case the total mass is de¬ned by μ = μ+ +

μ’ . We will also use the notation μ = |dμ|.

In the case where is a topological space, further concepts are usually needed.

De¬nition 5.32 Let be a topological space and O denote its collection of open sets.

The σ -algebra generated by O is called the Borel σ -algebra and denoted by B( ). If

is a Hausdorff space then a measure μ de¬ned on B( ) that satis¬es μ(K ) < ∞ for all

compact sets K ⊆ is called a Borel measure. The carrier of a Borel measure μ is the set

\ {U : U is open and μ(U ) = 0}.

Note that a Borel measure is more than just a measure de¬ned on Borel sets. The as-

sumption that is a Hausdorff space ensures that compact sets are closed and therefore

measurable. A ¬nite measure de¬ned on Borel sets is automatically a Borel measure.

If Q is a subset of a Hausdorff space then B(Q) is given by B(Q) = Q © B( ), using

the induced topology on Q.

In case of Rd , it is well known that B(Rd ) is also generated by the set of all semi-open

cubes [a, b) := {x ∈ Rd : a j ¤ x j < b j }. To be more precise, it is known that the set F d ,

which contains all ¬nite unions of such semi-open cubes, is a ring. Hence, any pre-measure

de¬ned on F d has an extension to B(Rd ).