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**Well-behaved convolution averages and the non-accumulation theorem for limit-cycles.**
*(English)*
Zbl 0857.34009

Braaksma, B. L. J. (ed.) et al., The Stokes phenomenon and Hilbert’s 16th problem. Proceedings of the workshop, Groningen, The Netherlands, May 31-June 3, 1995. Singapore: World Scientific. 71-101 (1996).

Summary: This expository paper introduces several uniformizing averages, which are serviceable in resummation theory because they manage to reconcile three essential, but at first sight quite conflicting demands: respecting convolution; preserving realness; reproducing latteral growth. Their potential range of application covers most situations characterized by a combination of (1) nonlinearity; (2) divergence, (3) realness. We sketch three typical applications, the last of which leads to a marginal, yet significant simplification of the constructive (i.e. resummation-theoretical) proof of the non-accumulation theorem for the limit-cycles of a real-analytic vector field on \(\mathbb{R}^2\).

For the entire collection see [Zbl 0846.00026].

For the entire collection see [Zbl 0846.00026].

### MSC:

34M99 | Ordinary differential equations in the complex domain |

32S05 | Local complex singularities |

32S65 | Singularities of holomorphic vector fields and foliations |

34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |

40A99 | Convergence and divergence of infinite limiting processes |

40H05 | Functional analytic methods in summability |

53B99 | Local differential geometry |

37-XX | Dynamical systems and ergodic theory |

60G15 | Gaussian processes |

60J60 | Diffusion processes |

60J65 | Brownian motion |