A regularity class for the roots of non-negative functions

We investigate the regularity of the positive roots of a non-negative function of one-variable. A modified H"older space ${\mathcal{F}}^{\beta }$ is introduced such that if $f\in {\mathcal{F}}^{\beta }$ then $f^\alpha \in C^{\alpha \beta}$. This provides sufficient conditions to overcome the usual limitation in the square root case ($\alpha =1\mathrm{/}2$) for H"older functions that $f^{1\mathrm{/}2}$ need be no more than $C^1$ in general. We also derive bounds on the wavelet coefficients of $f^{\alpha }$, which provide a finer understanding of its local regularity.