A regularity class for the roots of nonnegative functions

Kolyan Michael Ray; Johannes Schmidt-Hieber

doi:10.1007/s10231-017-0655-2

A regularity class for the roots of nonnegative functions

Kolyan Michael Ray, Johannes Schmidt-Hieber

Leiden University

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Abstract

We investigate the regularity of the positive roots of a nonnegative function of one-variable. A modified Hölder space F^β is introduced such that if f ∈ F^β then f^α ∈ C^(αβ) . This provides sufficient conditions to overcome the usual limitation in the square root case (α = 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^α , which provide a finer understanding of its local regularity.

Original language	English
Pages (from-to)	2091-2103
Number of pages	13
Journal	Annali di Matematica Pura ed Applicata
Volume	196
Issue number	6
Early online date	27 Mar 2017
DOIs	https://doi.org/10.1007/s10231-017-0655-2
Publication status	E-pub ahead of print - 27 Mar 2017

Keywords

Regularity of roots
Holder spaces
Differential inequalities
Wavelets

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AB - We investigate the regularity of the positive roots of a nonnegative function of one-variable. A modified Hölder space F^β is introduced such that if f ∈ F^β then f^α ∈ C^(αβ) . This provides sufficient conditions to overcome the usual limitation in the square root case (α = 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^α , which provide a finer understanding of its local regularity.

KW - Regularity of roots

KW - Holder spaces

KW - Differential inequalities

KW - Wavelets

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DO - 10.1007/s10231-017-0655-2

M3 - Article

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SP - 2091

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JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

IS - 6

ER -

A regularity class for the roots of nonnegative functions

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