+ Cδ m ’ /2

e’ ω 2 /2

¤ ω 2 dω

2

√

2 ω 2 / 2m>δ

¤

for suf¬ciently large m. The case x = 0 follows immediately by replacing (· + x)

by

in the previous case.

The approximation process described in Theorem 5.20, item (4), is a well-known method

of approximating a function. It is sometimes also called approximation by molli¬cation

or regularization. Let us stay a little longer with this process. In particular we are now

interested in replacing the Gaussians by an arbitrary compactly supported C ∞ -function.

Moreover, we are interested in weaker forms of convergence.

Lemma 5.21 Suppose that f ∈ L p (Rd ), 1 ¤ p < ∞, is given. Then we have

limx’0 f ’ f (· + x) L p (Rd ) = 0.

Proof Let us denote f (· + x) by f x . We start by showing the result for a continuous

function g with compact support. Choose a compact set K ⊆ Rd such that the support of

gx is contained in K for all x ∈ Rd with x 2 ¤ 1. Since g is continuous, it is uniformly

continuous on K . Hence for a given > 0 we ¬nd a δ > 0 such that |g(y) ’ g(x + y)| <

for all y ∈ K and all x 2 < δ. This ¬nishes the proof in this case because

g ’ gx = g ’ gx ¤ [vol(K )]1/ p .

L p (Rd ) L p (K )

Now, for an arbitrary f ∈ L p (Rd ) and > 0 we choose a function g ∈ C0 (Rd ) with

f ’ g L p (Rd ) < /3. By substitution this also means that f x ’ gx L p (Rd ) < /3, giving

f ’ fx ¤ f ’g + g ’ gx + gx ’ f x

L p (Rd ) L p (Rd ) L p (Rd ) L p (Rd )

2

< + g ’ gx L p (Rd ) ,

3

and this becomes smaller than for suf¬ciently small x 2 .

Note that the result is wrong in the case p = ∞. In fact, f ’ f (· ’ x) L ∞ (Rd ) ’ 0 as

x ’ 0 implies that f is almost everywhere uniformly continuous.

The previous result allows us to establish the convergence of approximation by convolu-

∞

tion, not only pointwise but also in L p . The following results are formulated for g ∈ C0 (Rd ),

but the proofs show that for example the second and third items hold even if we have only

g ∈ L 1 (Rd ).

58 Auxiliary tools from analysis and measure theory

∞

Theorem 5.22 Suppose an even and nonnegative g ∈ C0 (Rd ) is given, normed by

g(x)d x = 1. De¬ne gm (x) = m g(mx). Then the following are true.

d

∈ L loc (Rd ) then f — g ∈ C ∞ (Rd ) and D ± ( f — g) = f — (D ± g).

(1) If f 1

∈ L p (Rd ) with 1 ¤ p ¤ ∞ then f — g ∈ L p (Rd ) and f — g L p (R d ) ¤ f

(2) If f g L 1 (R d ) .

L p (R d )

∈ L p (Rd ) with 1 ¤ p < ∞ then f ’ f — gn L p (R d ) ’ 0 for n ’ ∞.

(3) If f

∈ C(Rd ) then f — gn ’ f uniformly on every compact subset of Rd .

(4) If f

Proof The ¬rst property is an immediate consequence of the theory of integrals depending

on an additional parameter. The second property is obviously true for p = 1, ∞. Moreover,

if 1 < p < ∞ we obtain, with q = p/( p ’ 1),

| f — g(x)| ¤ | f (y)||g(x ’ y)|1/ p |g(x ’ y)|1/q dy

Rd

1/ p 1/q

¤ | f (y)| |g(x ’ y)|dy |g(x ’ y)|dy

p

Rd Rd

or in other words

p/q

p

f —g ¤ |g(y)|dy | f (y)| p |g(x ’ y)|d yd x

L p (Rd )

Rd Rd Rd

p p

=g .

f

L 1 (Rd ) L p (Rd )

Using Minkowski™s inequality again, we can derive

p p

f ’ f — gn ¤ f ’ f (· + y/n) g(y)dy

L p (Rd ) L p (Rd )

Rd

in the same fashion as before. But since f ’ f (· + y/n) L p (Rd ) ’ 0 for n ’ ∞ by

Lemma 5.21 and since f ’ f (· + y/n) L p (Rd ) ¤ 2 f L p (Rd ) , Lebesgue™s dominated con-

vergence theorem yields the third property.

Finally, let f be a continuous function and > 0 be given. If K ⊆ Rd is a compact

set then we have K ⊆ B(0, R) for R > 0. Choose δ > 0 such that | f (x) ’ f (y)| < for

all x, y ∈ B(0, R + 1) with x ’ y 2 < δ. Without restriction we can assume that g is

supported in B(0, 1). This allows us to conclude, for x ∈ K , that

| f (x) ’ f — gn (x)| ¤ | f (x) ’ f (y)|gn (x ’ y)dy ¤

B(x,1/n)

whenever 1/n < δ, which establishes uniform convergence on K .

We now come back to the initial form of approximation by convolution, as given in

Theorem 5.20, to prove one of the fundamental theorems in Fourier analysis.

Theorem 5.23 The Fourier transform de¬nes an automorphism on S. The inverse mapping

is given by the inverse Fourier transformation. Furthermore, the L 2 (Rd )-norms of a function

and its transform coincide: f L 2 (Rd ) = f L 2 (Rd ) .

5.2 Fourier transform and approximation by convolution 59

Proof From Theorem 5.16 we can conclude that Fourier transformation maps S back

into S and that the same is valid for the inverse transformation. Using Theorem 5.16 and

Theorem 5.20 we obtain, again using Lebesgue™s convergence theorem,

f (x) = lim f (ω)gm (ω ’ x)dω

m’∞ Rd

= lim f (ω + x)g m (ω)dω

m’∞ Rd

ω

T

= lim f (ω)ei x gm (ω)dω

m’∞ Rd

= (2π )’d/2 f (ω)ei x ω dω.

T

Rd

Finally, we now have for an arbitrary g ∈ S,

g(x) = (2π )’d/2 g(ω)e’i x ω dω = (2π )’d/2

T

g(ω)ei x T ω dω = g(x),

Rd Rd

which, together with Theorem 5.16, leads to

f (x)g(x)d x = f (x)g(x)d x = f (x)g(x)d x,

Rd Rd Rd