Extension of \(C^\infty\) functions in polynomially bounded o-minimal structure (Q2238127)

In this paper, the author shows that if an o-minimal and polynomially bounded structure satisfies the condition (C) : ``for some \(n \ge 1\), any definable \(C^{\infty}\) function on \(\mathbb R^n\times\mathbb R_+\) admits a \(C^{\infty}\) definable extension on some \(\mathbb R^n\times (-\varepsilon,+\infty)\), with \(\varepsilon >0\)'', then any definable \(C^{\infty}\) function is analytic. The proof is mainly based on the following interesting argument that can already be pulled out from [\textit{A. Parusiński} and \textit{J.-P. Rolin}, Can. Math. Bull. 57, No. 3, 614--620 (2014; Zbl 1303.14067)]. Given a formal series in one variable \(f\in\mathbb R[[X]]\), write \(f(x+iy)=g(x,y)+ih(x,y)\) with \(g,h\in\mathbb R[[X,Y]]\). If \(g\) or \(h\) is the Taylor expansion of a \(C^{\infty}\) function which belongs to a quasi-analytic class closed under derivation, then it is analytic. The principle here is to mix the extension condition (C) together with definable operations to realize these real and imaginary part \(g\) and \(h\) from a definable function with Taylor expansion \(f\).