Let \(\mu\) be a measure on \((\mathbb{R}, \mathscr{B}(\mathbb{R})\) ), let \(u: \mathbb{R} ightarrow \mathbb{C}\) be a measurable function (see Problem 10.9 ) and denote by \(d x\) one-dimensional Lebesgue measure. Find conditions on \(\mu\) and \(u\) which guarantee that the so-called Fourier transforms

\[\widehat{\mu}(\xi):=\frac{1}{2 \pi} \int_{\mathbb{R}} e^{-i x \xi} \mu(d x) \quad \text { and } \quad \widehat{u}(\xi):=\frac{1}{2 \pi} \int_{\mathbb{R}} e^{-i x \xi} u(x) d x\]

exist resp. are continuous resp. are \(n\) times differentiable.