We investigate the regularity of functions such that , where is a given nonnegative function of one variable. Assuming that is of class () and vanishes together with its derivatives up to order at all its local minimum points, one can find a of class . Under the same assumption on the minimum points, if is of class then can be chosen such that it admits a derivative of order everywhere. Counterexamples show that these results are sharp.
@article{ASNSP_2010_5_9_3_635_0,
author = {Bony, Jean-Michel and Colombini, Ferruccio and Pernazza, Ludovico},
title = {On square roots of class $C^m$ of nonnegative functions of one variable},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {635--644},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {3},
year = {2010},
mrnumber = {2722658},
zbl = {1207.26004},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_3_635_0/}
}
TY - JOUR AU - Bony, Jean-Michel AU - Colombini, Ferruccio AU - Pernazza, Ludovico TI - On square roots of class $C^m$ of nonnegative functions of one variable JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 635 EP - 644 VL - 9 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2010_5_9_3_635_0/ LA - en ID - ASNSP_2010_5_9_3_635_0 ER -
%0 Journal Article %A Bony, Jean-Michel %A Colombini, Ferruccio %A Pernazza, Ludovico %T On square roots of class $C^m$ of nonnegative functions of one variable %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 635-644 %V 9 %N 3 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2010_5_9_3_635_0/ %G en %F ASNSP_2010_5_9_3_635_0
Bony, Jean-Michel; Colombini, Ferruccio; Pernazza, Ludovico. On square roots of class $C^m$ of nonnegative functions of one variable. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 635-644. http://www.numdam.org/item/ASNSP_2010_5_9_3_635_0/
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