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}, zbl = {1207.26004}, mrnumber = {2722658}, language = {en}, url = {archive.numdam.org/item/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/
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