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arXiv · 1510.09054

A regularity class for the roots of non-negative functions

Kolyan Ray·Johannes Schmidt-Hieber
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Abstract

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 ($\alpha = 1/2$) for H\"older functions that $f^{1/2}$ 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.

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Kolyan Ray, Johannes Schmidt-Hieber. 2015-10-30. A regularity class for the roots of non-negative functions. https://doi.org/10.1007/s10231-017-0655-2

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