Measure, Integration & Real Analysis · Problem 11C.1

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · fourier-transform-norm

Problem

Suppose

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

. Prove that

\|\hat{f}\|_\infty = \|f\|_1

if and only if there exists

\zeta \in \partial \mathbf{D}

and

t \in \mathbf{R}

such that

\zeta f(x) e^{-itx} \geq 0

for almost every

x \in \mathbf{R}