1108 CHAPTER 32. FOURIER TRANSFORMS

Also by Theorem 32.3.12 {Fφ k}∞

k=1 is Cauchy in L2 (Rn) and so it converges to someh ∈ L2 (Rn). Therefore, from the above,

F f (ψ) =∫Rn

h(x)ψ (x)

which shows that F ( f ) ∈ L2 (Rn) and h = F ( f ) . The case of F−1 is entirely similar.Since F f and F−1 f are in L2 (Rn) , this also proves the following theorem.

Theorem 32.3.14 If f ∈ L2(Rn), F f and F−1 f are the unique elements of L2 (Rn) suchthat for all φ ∈ G , ∫

RnF f (x)φ(x)dx =

∫Rn

f (x)Fφ(x)dx, (32.3.8)∫Rn

F−1 f (x)φ(x)dx =∫Rn

f (x)F−1φ(x)dx. (32.3.9)

Theorem 32.3.15 (Plancherel)

|| f ||2 = ||F f ||2 = ||F−1 f ||2. (32.3.10)

Proof: Use the density of G in L2 (Rn) to obtain a sequence, {φ k} converging to f inL2 (Rn). Then by Lemma 32.3.13

||F f ||2 = limk→∞

||Fφ k||2 = limk→∞

||φ k||2 = || f ||2 .

Similarly,|| f ||2 = ||F−1 f ||2.

The following corollary is a simple generalization of this. To prove this corollary,use the following simple lemma which comes as a consequence of the Cauchy Schwarzinequality.

Lemma 32.3.16 Suppose fk→ f in L2 (Rn) and gk→ g in L2 (Rn). Then

limk→∞

∫Rn

fkgkdx =∫Rn

f gdx

Proof: ∣∣∣∣∫Rnfkgkdx−

∫Rn

f gdx∣∣∣∣≤ ∣∣∣∣∫Rn

fkgkdx−∫Rn

fkgdx∣∣∣∣+∣∣∣∣∫Rn

fkgdx−∫Rn

f gdx∣∣∣∣

≤ || fk||2 ||g−gk||2 + ||g||2 || fk− f ||2 .

Now || fk||2 is a Cauchy sequence and so it is bounded independent of k. Therefore, theabove expression is smaller than ε whenever k is large enough.