Let $\mu$ be a (complex) measure on the torus $[0,1]$. Then we can define its Fourier coefficients $\hat \mu(k)=\int_0^1 e^{-2\pi i k x}dx$ for $k\in \mathbb Z$. We can also define its continuous Fourier transform $\hat \mu(\xi)=\int_0^1 e^{-2\pi i \xi x}dx$ for $\xi\in \mathbb R$.
One theorem (A) states that $\mu (\xi)=O(|\xi|^{-\alpha})$ if and only if $\mu (k)=O(|k|^{-\alpha})$. It refers to Lemma 9.A.4 in Wolff's harmonic analysis notes (can be found on Page 63 of http://www.math.ubc.ca/~ilaba/wolff/notes_march2002.pdf), which is a slightly different version. So I would like to find a more direct reference for theorem (A).