Measure, Integration & Real Analysis · Problem 11C.14

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · fourier-transform-conjugate

Problem

Suppose

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

. (a) Prove that

\widehat{\bar{f}}(t) = \overline{\hat{f}(-t)}

for all

t \in \mathbf{R}

. (b) Prove that

f(x) \in \mathbf{R}

for almost every

x \in \mathbf{R}

if and only if

\hat{f}(t) = \overline{\hat{f}(-t)}

for all

t \in \mathbf{R}