On the existence of holomorphic functions having prescribed asymptotic expansions.

*(English)*Zbl 0816.46019A holomorphic function \(f\) on a domain \(\Omega\in\mathbb{C}\) has an asymptotic expansion at \(u\in \partial\Omega\) if the limits \(a_ 0= \lim_{z\to u} f(z)\) and, for every \(n\in \mathbb{N}\),
\[
a_ n= \lim_{z\to u} (z- u)^{- n} \left(f(z)- \sum^{n-1}_{j= 0} a_ j(z- u)^ j\right)
\]
exist (and are finite); then \(\sum^ \infty_{n= 0} a_ n(z- u)^ n\) is called the asymptotic expansion of \(f\) at \(u\). For \(D\subset \partial\Omega\), \(A(\Omega; D)\) denotes the space of all holomorphic functions \(f\) on \(\Omega\) which have an asymptotic expansion at every point \(u\in D\). \(D\) is said to be regularly asymptotic for \(\Omega\) if, for every family \(\{c_{u,n}; u\in D, n\in \mathbb{N}_ 0\}\) of complex numbers, there is \(f\in A(\Omega; D)\) with the asymptotic expansion \(\sum^ \infty_{n= 0} c_{u, n} (z- u)^ n\) at \(u\) for every \(u\in D\).

If \(D\) is regularly asymptotic for \(\Omega\), then it has no accumulation point (and hence is at most countable). Long ago, T. Carleman showed that every finite subset \(D\) of the boundary of a bounded open and convex set \(\Omega\subset \mathbb{C}\) is regularly asymptotic for \(\Omega\). Here the following theorem, generalizing Carleman’s result and some previous results of the second-named author, is proved:

If \(D\) is a non-empty subset of \(\partial\Omega\) having no accumulation point and if \(\partial\Omega\) is “quasi-connected” at every point of \(D\), then \(D\) is regularly asymptotic for \(\Omega\). For a “practical form”, the notion of quasi-connectedness is analyzed. It follows that \(D\subset \partial\Omega\) without accumulation point is regularly asymptotic for \(\Omega\) whenever the connected component of every point of \(D\) in \(\partial\Omega\) has more than one point; if \(\Omega\) is simply connected, \(\partial\Omega\) is quasi-connected at every point of \(\partial\Omega\). The proofs use properties of Fréchet spaces and of quasi-LB representations, cf. [the second author, J. Lond. Math. Soc., II. Ser. 35, 149-168 (1987; Zbl 0625.46006)].

If \(D\) is regularly asymptotic for \(\Omega\), then it has no accumulation point (and hence is at most countable). Long ago, T. Carleman showed that every finite subset \(D\) of the boundary of a bounded open and convex set \(\Omega\subset \mathbb{C}\) is regularly asymptotic for \(\Omega\). Here the following theorem, generalizing Carleman’s result and some previous results of the second-named author, is proved:

If \(D\) is a non-empty subset of \(\partial\Omega\) having no accumulation point and if \(\partial\Omega\) is “quasi-connected” at every point of \(D\), then \(D\) is regularly asymptotic for \(\Omega\). For a “practical form”, the notion of quasi-connectedness is analyzed. It follows that \(D\subset \partial\Omega\) without accumulation point is regularly asymptotic for \(\Omega\) whenever the connected component of every point of \(D\) in \(\partial\Omega\) has more than one point; if \(\Omega\) is simply connected, \(\partial\Omega\) is quasi-connected at every point of \(\partial\Omega\). The proofs use properties of Fréchet spaces and of quasi-LB representations, cf. [the second author, J. Lond. Math. Soc., II. Ser. 35, 149-168 (1987; Zbl 0625.46006)].

Reviewer: K.D.Bierstedt (Paderborn)

##### MSC:

46E10 | Topological linear spaces of continuous, differentiable or analytic functions |

30B99 | Series expansions of functions of one complex variable |

46A04 | Locally convex Fréchet spaces and (DF)-spaces |

46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |

30E15 | Asymptotic representations in the complex plane |

46A11 | Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) |