A regularity class for the roots of nonnegative functions

• DOI: 10.1007/s10231-017-0655-2
• Publication Date: 2017-03-27T07:21:31Z
• Authors: Kolyan Ray; Johannes Schmidt-...
• Keywords: Regularity of roots; Mathematics - Functional Analysis; Differential inequalities; Holder spaces; Applied Mathematics; FOS: Mathematics; Wavelets; 26A16, 26A27; 510; Functional Analysis (math.FA)

We investigate the regularity of the positive roots of a non-negative function of one-variable. A modified H��lder space $\mathcal{F}^��$ is introduced such that if $f\in \mathcal{F}^��$ then $f^��\in C^{����}$. This provides sufficient conditions to overcome the usual limitation in the square root case ($��= 1/2$) for H��lder functions that $f^{1/2}$ need be no more than $C^1$ in general. We also derive bounds on the wavelet coefficients of $f^��$, which provide a finer understanding of its local regularity.<br/>12 pages<br/>