Holder continuity of Fourier transform of measure

Question

I am trying to prove the following, say $\mu$ is a non-negative Borel measure on $\mathbb{R}$ with $\mu(\mathbb{R}) = 1$ and $$ f(t) = \widehat{\mu}(t) = \int_{\mathbb{R}} e^{itx}\;d\mu(x).$$ Assuming that $$\int_{\mathbb{R}} |x|^{2+\delta}\;d\mu(x) < \infty$$ Can we show that $f\in \mathrm{C}^{2,\delta}(\mathbb{R})$?