Borel theorem

2020 Mathematics Subject Classification: Primary: 26E10,34E05 Secondary: 30E15 [MSN][ZBL]

A class of theorems guaranteeing existence of a smooth function with any preassigned (eventually diverging) Taylor series, including statements for complex functions defined in sectorial domains.

Real version

For any collection of real numbers $\{c_\alpha:\ \alpha\in\Z_+^n\}$ labeled by multiindices there exists a $C^\infty$-smooth function $f:(\R^n,0)\to\R$ such that $c_\alpha=\frac1{\alpha!}\partial^\alpha f(0)$. In other words, any formal series $\sum_{|\alpha|\ge 0} c_\alpha x^\alpha\in\R[[x_1,\dots,x_n]]$ is the Taylor series of a $C^\infty$-smooth function defined in an open neighborhood of the origin.

In this form the Borel theorem is a particular case of the Whitney extension theorem, see [N].

Complex version

Let $S\subset(\C,0)$ be an open sector $\{0<|z|<\rho,\ |\theta_-<\arg z <\theta_+\}$ with the opening angle $\theta_+-\theta_-$ less than $2\pi$ on the complex plane with the vertex at the origin, and $\{c_k:\ k=0,1,2,\dots\}$ a sequence of complex numbers

Then there exists a function $f$ holomorphic in $S$, for which the formal series $\sum c_k z^k$ is an asymptotic series: $$ \forall m\in\N\quad \lim_{z\to 0} \frac1{z^m}\Big(f(z)-\sum_1^m c_k z^k\Big)=0\qquad\text{as }z\to 0,\ z\in S. $$ This theorem is also referred to as the Borel-Ritt theorem, see [W, Sect. 9].

Remark

One can consider also sectors with opening larger than $2\pi$, but only on the suitable Riemann surface. A (single-valued) function defined in a punctured neighborhood of the origin and admitting an asymptotic series, is necessarily holomorphic, so its asymptotic series must converge.

Multidimensional version

The analog of Borel-Ritt theorem is valid also for formal series in several variables: any such series can be realized as an asymptotic series for a suitable function of several complex variables, holomorphic in a proper polysector $z\in\{\C^n:\ \theta_{i-}<\arg z_i<\theta_{i+},\ i=1,\dots,n,\ 0<|z|<\rho\}$, provided that $\theta_{i+}-\theta_{i-}<2\pi$. See [R].

References

[W]	Wasow, W. Asymptotic expansions for ordinary differential equations, Dover Publications, Inc., New York, 1987. MR0919406
[N]	Narasimhan, R. Analysis on real and complex manifolds, North-Holland Mathematical Library, 35. North-Holland Publishing Co., Amsterdam, 1985. MR0832683
[R]	Ramis, J.-P. À propos du théorème de Borel-Ritt à plusieurs variables, Lecture Notes in Math., 712, Équations différentielles et systèmes de Pfaff dans le champ complexe (Sem., Inst. Rech. Math. Avancée, Strasbourg, 1975), pp. 289--292, Springer, Berlin, 1979. MR0548148