Analytic extension through a T-filter. (Prolongement analytique à travers un T-filtre.) (French) Zbl 0596.32007

Sarmant, Marie-Claude; Escassut, Alain

Stud. Sci. Math. Hung. 22, No. 1-4, 407-444 (1987).

Summary: Let \((K, \vert\cdot\vert)\) be an algebraically closed complete ultrametric field. Let \(D\) be a closed bounded infraconnected subset of \(K\) provided with a T-filter \(\mathcal F\) with empty beach, let \(H(D)\) be the Banach algebra of the analytic elements on \(D\), and let \(\mathcal T(\mathcal F)\) be the ideal of the \(f\in H(D)\) that approach \(0\) on \(\mathcal F\). If all the holes of \(D\) (or at least those of a T-sequence associated to \(\mathcal F)\) have diameters lower bounded by a same \(\rho >0\), then the algebra \(A=H(D)/\mathcal T(\mathcal F)\) is complete for the absolute value defined by \(\mathcal F(| \bar f| =\lim_{\mathcal F} \vert f(x)\vert\) with \(\bar f\) the image of \(f\) in \(A)\).
Now suppose the beach \(\mathcal P(\mathcal F)\) of \(\mathcal F\) is not empty. Under the same hypothesis on the diameters of holes, we prove any \(f\in H(\mathcal P(\mathcal F))\) extends to an element of \(H(D)\) (in an infinity of ways). For instance let \(f\) be a Taylor series converging for \(\vert x\vert\leq R\), let \(b_n\) be a sequence in \(K\) such that \(\vert b_n\vert > \vert b_{n+1}\vert\), \(\lim \vert b_n| = R\), \(\prod^{\infty}_{n=1} R/ \vert b_n\vert = 0\) and for any \(\rho >0\) let \(\Lambda_{\rho} = K\setminus (\cup^{\infty}_{n=1}d^-(b_n,\rho)).\) Then \(f\) does extend to an analytic element on \(\Lambda_{\rho}\) meromorphic in \(K\), with a simple pole at each \(b_n\).

MSC:

32P05	Non-Archimedean analysis
32D99	Analytic continuation
12J25	Non-Archimedean valued fields

Keywords:

bounded infraconnected subset of complete ultrametric field; Banach algebra; meromorphic continuation