A regularity class for the roots of non-negative functions

Abstract

We investigate the regularity of the positive roots of a non-negative function of one-variable. A modified Hölder 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 Hölder 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.

12 pages

References in corpus (4)

• Adaptation to lowest density regions with application to support recovery
• Regularity of roots of polynomials
• A new proof of Bronshtein's theorem
• Minimax theory for a class of non-linear statistical inverse problems

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