Extend Plancherel's theorem (Theorem 19.20 ) to show that

\[\int \widehat{u}(\xi) \overline{\widehat{v}(\xi)} d \xi=(2 \pi)^{-n} \int u(x) \overline{v(x)} d x \quad \forall u, v \in L_{\mathbb{C}}^{2}\left(\lambda^{n}ight)\]

Data from theorem 19.20

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Theorem 19.20 (Plancherel) If u = L(X") L(X"), then ||||2 = (2)-/2||u|| 2. In particular, there is a continuous extension F: L(X") L(X"). Proof Let u A(R"). Since u = L(X") L(X") and = L(X"), we see from Theorem 19.12 [ \(&) d = [(E)(E)d = 19.9 (2)" [()(E)d (27)" [u(x) F[] (x) dx (27)" [|u(x)| dx. Since the Wiener algebra is dense in L(X"), see Lemma 19.19, we can approx- imate any u L(R") with a sequence (uk) k CA(R"). Therefore, ||FUK - FUm||2 = (2)-n/2 ||uk Um||2 - - (19.14) 0. k,mo This shows that (Fuk) kEN is a Cauchy sequence in L(X"). Because of the completeness, the limit L-lim-o Fuk exists (and it is independent of the