Measure, Integration & Real Analysis · Problem 11C.9

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · fourier-transform-nth-derivative

Problem

Suppose

n \in \mathbf{Z}^+

and

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

n

times continuously differentiable and

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

for

k = 1, \ldots, n

. Prove that if

t \in \mathbf{R}

, then

\widehat{f^{(n)}}(t) = (2\pi it)^n \hat{f}(t).