32.2. FOURIER TRANSFORMS OF FUNCTIONS IN G 1099

Lemma 32.2.2 The following formulas are true. (c > 0)∫R

e−ct2e−istdt =

∫R

e−ct2eistdt = e−

s24c

√π√c, (32.2.1)

∫Rn

e−c|t|2e−is·tdt =∫Rn

e−c|t|2eis·tdt = e−|s|24c

(√π√c

)n

. (32.2.2)

Proof: Consider the first one. Let h(s) be given by the left side. Then

H (s)≡∫R

e−ct2e−istdt =

∫R

e−ct2cos(st)dt

Then using the dominated convergence theorem to differentiate,

H ′ (s) =∫R−e−ct2

t sin(st)dt =e−ct2

2csin(st) |∞−∞−

s2c

∫R

e−ct2cos(st)dt =− s

2cH (s) .

Also H (0) =∫R e−ct2

dt. Thus H (0) =∫R e−cx2

dx≡ I and so

I2 =∫R2

e−c(x2+y2)dxdy =∫

∞

0

∫ 2π

0e−cr2

rdθdr =π

c.

Hence

H ′ (s)+s

2cH (s) = 0, H (0) =

√π

c.

It follows that H (s) = e−s24c

√π√c . The second formula follows right away from Fubini’s

theorem.With these formulas, it is easy to verify F,F−1 map G to G and F ◦F−1 = F−1 ◦F = id.

Theorem 32.2.3 Each of F and F−1 map G to G . Also F−1◦F (ψ)=ψ and F ◦F−1 (ψ)=ψ .

Proof: The first claim will be shown if it is shown that Fψ ∈ G for ψ (x) = xα e−b|x|2

because an arbitrary function of G is a finite sum of scalar multiples of functions such asψ . Using Lemma 32.2.2,

Fψ (t) ≡(

12π

)n/2 ∫Rn

e−it·xxα e−b|x|2dx

=

(1

2π

)n/2

(i)−|α|Dαt

(∫Rn

e−it·xe−b|x|2dx)

=

(1

2π

)n/2

(i)−|α|Dαt

(e−|t|24b

(√π√b

)n)

and this is clearly in G because it equals a polynomial times e−|t|24b .