Measure, Integration & Real Analysis · Problem 11C.8

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · fourier-transform-higher-derivatives

Problem

Suppose

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

and

n \in \mathbf{Z}^+

. Define

g: \mathbf{R} \to \mathbf{C}

g(x) = x^n f(x)

. Prove that if

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

, then

\hat{f}

n

times continuously differentiable on

\mathbf{R}

and

\left(\hat{f}\right)^{(n)}(t) = (-2\pi i)^n \hat{g}(t)

for all

t \in \mathbf{R}