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Sums of squares of derivable functions. (Sommes de carrés de fonctions dérivables.) (French) Zbl 1107.26008
It is proved that any positive smooth function can be written as the sum of two differentiable functions. The main result asserts that if \(I\subset\mathbb R\) is an interval and \(f:I\rightarrow\mathbb R\) is a positive function of class \(C^{2m}\) (\(m\geq 1\)), then \(f=g^2+h^2\), where \(g,h\in C^m(I)\). Next, the author proves that if \(\Omega\subset\mathbb R^2\) is an open set and \(f:\Omega\rightarrow\mathbb R\) is a positive function of class \(C^4\) such that \(\{ f(x)=\nabla^2f(x)=0\}\Rightarrow\nabla^4f(x)=0\), then there exists a positive integer \(N\) and functions \(g_i\) (\(1\leq i\leq N\)) such that \(f=\sum_{i=1}^Ng_i^2\).
It is also argued that \(N=78\) is convenient in the above result. An interesting question would be to find the least positive integer \(N\) such that this property remains true. The proofs rely on a careful local analysis of \(f\) combined with precise estimates of the function and its derivatives.

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26B05 Continuity and differentiation questions
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