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://archive.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://archive.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://archive.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://archive.numdam.org/item/ASNSP_2010_5_9_3_635_0/
[1] , , and , Choosing roots of polynomials smoothly, Israel J. Math. 105 (1998), 203–233. | MR | Zbl
[2] , Sommes de carrés de fonctions dérivables, Bull. Soc. Math. France 133 (2005), 619–639. | EuDML | Numdam | MR
[3] , , and , Nonnegative functions as squares or sums of squares, J. Funct. Anal. 232 (2006), 137–147. | MR | Zbl
[4] , and , On the differentiability class of the admissible square roots of regular nonnegative functions, In: “Phase Space Analysis of Partial Differential Equations”, 45–53, Progr. Nonlinear Differential Equations Appl., Vol. 69, Birkhäuser Boston, Boston, MA, 2006. | MR | Zbl
[5] , Note sur une nouvelle formule du calcul différentiel, Quarterly J. Pure Appl. Math. 1 (1857), 359–360.
[6] , Racine carrée d’une fonction différentiable, Ann. Inst. Fourier (Grenoble) 13 (1963), 203–210. | EuDML | Numdam | MR | Zbl
[7] , and , Choosing roots of polynomials smoothly, II, Israel J. Math. 139 (2004), 183–188. | MR | Zbl
[8] , Smoothness of roots of hyperbolic polynomials with respect to one-dimensional parameter, Bull. Fac. Gen. Ed. Gifu Univ. 21 (1985), 115–118. | MR
