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Nonnegative functions as squares or sums of squares. (English) Zbl 1093.26007
In 1978 Fefferman and Phong stated (and sketchily proved) a lemma that any nonnegative $${\mathcal C}^{\infty}$$ function in $${\mathbb R}^n$$ is a sum of squares $${\mathcal C}^{1,1}$$. In the reviewed paper it is proved for $$n\geq 4$$ that such a regularity condition is sharp, namely, there exist nonnegative $${\mathcal C}^{\infty}$$ functions $$f : {\mathbb R}^n \rightarrow {\mathbb R}$$ that are not sums of squares of $${\mathcal C}^{2}$$ functions. It is also proved that a general nonnegative $${\mathcal C}^{2}$$ function of one variable has a $${\mathcal C}^{1}$$ admissible square root, but no better regularity can be assured. If the function is $${\mathcal C}^{4}$$ and its values at all its local minima are controlled it has a $${\mathcal C}^{2}$$ admissible square root, but no better regularity can be assured. In both cases, increasing the regularity of the nonnegative function up to $${\mathcal C}^{\infty}$$ does not provide a better result.

##### MSC:
 26B40 Representation and superposition of functions 26C99 Polynomials, rational functions in real analysis 26E10 $$C^\infty$$-functions, quasi-analytic functions
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##### References:
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