0

Then f possesses a generalized Fourier transform f of order /2 with ω ’ f (ω) ω 2 ∈

L 2 (Rd ). This means that the function f q± is in L 2 (Rd ) for all |±| = . Hence we can de¬ne

f ± (ω) := ( f q± (i·))∨ (ω) ∈ L 2 (Rd )

for |±| = , using the inverse L 2 -Fourier transform. Since f is real-valued, so is f ± . Since

∞

q± γ ∨ ∈ S for γ ∈ C0 (Rd ) we ¬nd that

( f q± (i·))∨ (ω)γ (ω)dω

f ± (ω)γ (ω)dω =

Rd Rd

f (ω)(iω)± γ ∨ (ω)dω

=

Rd

= (’1)|±| f (ω)(D ± γ )∨ (ω)dω

Rd

= (’1)|±| f (x)D ± γ (x)d x,

Rd

showing that f ± is the generalized derivative of f . Hence, the native space is contained in

the Beppo Levi space.

Finally, for f, g ∈ N d, (Rd ) we have

!

( f, g)BL (Rd ) = f ± (x)g± (x)d x

±! Rd

|±|=

!

= f ± (ω)g± (ω)dω

±! Rd

|±|=

!

f (ω)g(ω)(iω)± (’iω)± dω

=

±! Rd

|±|=

= f (ω)g(ω) ω 2

dω

2

Rd

f (ω)g(ω)

= (2π)’d/2 dω

d, (ω)

Rd

= ( f, g)N (Rd ) .

d,

It remains to show that the inclusion is actually an identity. One might be tempted to

de¬ne a generalized Fourier transform f for a function f from the Beppo Levi space by

f ± (ω)

.

f (ω) :=

(iω)±

164 Native spaces

Unfortunately, it is not at all simple to prove that this is indeed the generalized Fourier

transform in our sense. The reason for this is that γ (ω)/(iω)± is not even continuous at zero

for γ ∈ S .

Hence, instead of proving that the Beppo Levi space is a subspace of the native space we

will show that every function from the Beppo Levi space that is orthogonal to all functions

from the native space with respect to the semi-inner product of the Beppo Levi space is

actually a polynomial of degree less than .

The ¬rst step in this direction is to reformulate Theorem 10.36 in a way appropriate for

Beppo Levi functions. To this end we want to employ density results. But since the Beppo

Levi space is only equipped with a semi-inner product, we have to be more precise about

what we understand by density.

∞

Theorem 10.40 Let > d/2. Then the set C0 (Rd ) is dense in BL (Rd ). To be more precise,

for every f ∈ BL (Rd ), every compact subset K of Rd , and every > 0 there exists a

∞

function g ∈ C0 (Rd ) such that

f ’ g L ∞ (K ) < ,

(1)

D ± f ’ D ± g L 2 (R d ) < for all |±| = .

(2)

Proof In the ¬rst step we will show that the set C ∞ (Rd ) © BL (Rd ) is dense in BL (Rd )

in the sense speci¬ed in the theorem. This follows immediately from approximation by

convolution. If we use f — gn , where {gn } is a sequence from Theorem 5.22 then we know

from this theorem that f — gn ∈ C ∞ (Rd ) and that property (1) is satis¬ed for suf¬ciently

large n. Moreover, since D ± ( f — gn ) = (D ± f ) — gn and D ± f ∈ L 2 (Rd ) for |±| = , the

same theorem tells us that f — gn ∈ BL (Rd ) and that the second property is also satis¬ed

for suf¬ciently large n.

Hence, it remains to show that the functions of C ∞ (Rd ) © BL (Rd ) can be approximated

∞

by C0 (Rd ) functions in the stated way. Let us assume that f ∈ C ∞ (Rd ) © BL (Rd ) is

∞

given. We choose a function ψ ∈ C0 (Rd ), which is identically one on x 2 ¤ 1, identically

∞

zero on x 2 ≥ 2, and has maximum absolute value one, and set f k := ψ(·/k) f ∈ C0 (Rd ).

Then Leibniz™ rule gives

±1β

D ± f k (x) = D ψ(x/k)D ±’β f (x) + ψ(x/k)D ± f (x),

|β|

βk

0=β¤±

so that

±

1

D± f ’ D± fk Dβ ψ D ±’β f

¤

L 2 (Rd ) L ∞ (Rd ) L 2 (Rd )

k 0=β¤± β

1/2

±

+ |D f (x)| d x .

2

2 >k

x

The last expression clearly tends to zero as k ’ ∞, which settles the second property,

while the ¬rst one is obvious.

10.5 Special cases of native spaces 165

This density result allows us to draw some very important conclusions.

Theorem 10.41 Suppose that » = N » j δx j is an element of L π ’1 (Rd ) , i.e. »( p) = 0

j=1

for all p ∈ π ’1 (Rd ). Set f » = » y d, (· ’ y) = » j d, (· ’ x j ). Then for every f ∈

BL (Rd ), > d/2, and every x ∈ Rd we have the representation

N

» j f (x ’ x j ) = ( f, f » (x ’ ·))BL (Rd ) (10.13)

j=1

!

D ± f (y)D ± f » (x ’ y)dy.

=

±! Rd

|±|=

Proof One consequence of Proposition 10.39 is that f » , which is an element of the native

space, is contained in the Beppo Levi space. Hence D ± f » is an element of L 2 (Rd ). Since

also D ± f ∈ L 2 (Rd ) by de¬nition, it follows easily from Lemma 5.21 that the right-hand

side of (10.13) is a continuous function.

∞

Next, let us ¬rst assume that f ∈ C0 (Rd ). Then the de¬nition of generalized derivatives,

the choice of coef¬cients in the semi-inner product, and Theorem 10.36 give

!

D ± f (x ’ y)D ± f » (y)dy