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, p. 635-644
We investigate the regularity of functions g such that g 2 =f, where f is a given nonnegative function of one variable. Assuming that f is of class C 2m (m>1) and vanishes together with its derivatives up to order 2m-4 at all its local minimum points, one can find a g of class C m . Under the same assumption on the minimum points, if f is of class C 2m+2 then g can be chosen such that it admits a derivative of order m+1 everywhere. Counterexamples show that these results are sharp.
Classification:  26A15,  26A27
@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},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {3},
     year = {2010},
     pages = {635-644},
     zbl = {1207.26004},
     mrnumber = {2722658},
     language = {en},
     url = {http://www.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://www.numdam.org/item/ASNSP_2010_5_9_3_635_0/

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