1104 CHAPTER 32. FOURIER TRANSFORMS

Theorem 32.3.8 Let f be a measurable function with polynomial growth,

| f (x)| ≤C(

1+ |x|2)N

for some N,

or let f ∈ Lp (Rn) for some p ∈ [1,∞]. Then f ∈ G ∗ if

f (φ)≡∫

f φdx.

Proof: Let f have polynomial growth first. Then the above integral is clearly welldefined and so in this case, f ∈ G ∗.

Next suppose f ∈ Lp (Rn) with ∞ > p≥ 1. Then it is clear again that the above integralis well defined because of the fact that φ is a sum of polynomials times exponentials of theform e−c|x|2 and these are in Lp′ (Rn). Also φ → f (φ) is clearly linear in both cases.

This has shown that for nearly any reasonable function, you can define its Fourier trans-form as described above. You could also define the Fourier transform of a finite Borelmeasure µ because for such a measure

ψ →∫Rn

ψdµ

is a linear functional on G . This includes the very important case of probability distributionmeasures. The theoretical basis for this assertion will be given a little later.

32.3.2 Fourier Transforms Of Functions In L1 (Rn)

First suppose f ∈ L1 (Rn) .

Theorem 32.3.9 Let f ∈ L1 (Rn) . Then F f (φ) =∫Rn gφdt where

g(t) =(

12π

)n/2 ∫Rn

e−it·x f (x)dx

and F−1 f (φ) =∫Rn gφdt where g(t) =

( 12π

)n/2 ∫Rn eit·x f (x)dx. In short,

F f (t)≡ (2π)−n/2∫Rn

e−it·x f (x)dx,

F−1 f (t)≡ (2π)−n/2∫Rn

eit·x f (x)dx.

Proof: From the definition and Fubini’s theorem,

F f (φ) ≡∫Rn

f (t)Fφ (t)dt =∫Rn

f (t)(

12π

)n/2 ∫Rn

e−it·xφ (x)dxdt

=∫Rn

((1

2π

)n/2 ∫Rn

f (t)e−it·xdt

)φ (x)dx.

Since φ ∈ G is arbitrary, it follows from Theorem 32.3.7 that F f (x) is given by the claimedformula. The case of F−1 is identical.

Here are interesting properties of these Fourier transforms of functions in L1.