A regularity class for the roots of non-negative functions

TL;DR

This paper introduces a new regularity class for non-negative functions that ensures their roots have higher regularity, overcoming typical limitations, and provides bounds on wavelet coefficients for detailed local regularity analysis.

Contribution

The paper defines a modified H"older space $^eta$ that guarantees higher regularity of roots of non-negative functions and derives bounds on wavelet coefficients for these roots.

Findings

If $f \\in \\mathcal{F}^\beta$, then $f^\alpha \\in C^{\alpha \beta}$.

Provides sufficient conditions for regularity of roots beyond the usual $C^1$ limit.

Establishes bounds on wavelet coefficients of $f^\alpha$.

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.