Let $\alpha > 0$ and define
\begin{equation*} \mathbb{H}^{\alpha}\left[-\pi,\pi\right] = \left\{f:\left[-\pi,\pi\right]\mapsto\mathbb{R} \;s.t.\; \sum\limits_{n\in\mathbb{Z}} \left\lvert n\right\lvert ^{\alpha}\left\lvert f_n\right\lvert < \infty \right\} \end{equation*}
In which $\left\{f_n\right\}^{\infty}_{n=-\infty}$ are associated Fourier coefficients of $f$. Is it true to say that if $f\in\mathbb{H}^{\alpha}\left[-\pi,\pi\right]$ and $f > 0$ almost sure (with respect to say Lebesgue measure), then $\frac{1}{f},\frac{1}{\sqrt{f}}\in\mathbb{H}^{\alpha}\left[-\pi,\pi\right]$?