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Vitali’s theorem without uniform boundedness. (English) Zbl 1347.41002

Summary: Let \(\{f_m\}_{m \geq 1}\) be a sequence of holomorphic functions defined on a bounded domain \(D \subset \mathbb{C}^n\) or a sequence of rational functions \((1 \leq \deg r_m \leq m)\) defined on \(\mathbb{C}^n\). We are interested in finding sufficient conditions to ensure the convergence of \(\{f_m\}_{m \geq 1}\) on a large set provided the convergence holds pointwise on a not too small set. This type of result is inspired from a theorem of Vitali which gives a positive answer for uniformly bounded sequence.

MSC:

41A05 Interpolation in approximation theory
41A63 Multidimensional problems
46A32 Spaces of linear operators; topological tensor products; approximation properties
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