Measure, Integration & Real Analysis · Problem 11C.18

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · plancherel-theorem-extension

Problem

Prove that if

f \in L^1(\mathbf{R})

, then

\|\hat{f}\|_2 = \|f\|_2

. [This exercise slightly improves Plancherel's Theorem (11.82) because here we have the weaker hypothesis that

f \in L^1(\mathbf{R})

instead of

f \in L^1(\mathbf{R}) \cap L^2(\mathbf{R})

. Because of Plancherel's Theorem, here you need only prove that if

f \in L^1(\mathbf{R})

and

\|f\|_2 = \infty

, then

\|\hat{f}\|_2 = \infty