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Vitali's theorem without uniform boundedness

Nguyen Quang Dieu, Phung Van Manh, Pham Hien Bang, and Le Thanh Hung

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Let $\{f_m\}_{m \ge 1}$ be a sequence of holomorphic functions defined on a bounded domain $D \subset \mathbb C^n$ or a sequence of rational functions $(1 \le \deg r_m \le m)$ defined on $\mathbb C^n$. We are interested in finding sufficient conditions to ensure the convergence of $\{f_m\}_{m \ge 1}$ 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.

Article information

Publ. Mat., Volume 60, Number 2 (2016), 311-334.

Received: 25 August 2014
Revised: 29 January 2015
First available in Project Euclid: 11 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 41A05: Interpolation [See also 42A15 and 65D05] 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section) 46A32: Spaces of linear operators; topological tensor products; approximation properties [See also 46B28, 46M05, 47L05, 47L20]

Rapid convergence convergence in capacity pluripolar set relative capacity


Dieu, Nguyen Quang; Manh, Phung Van; Bang, Pham Hien; Hung, Le Thanh. Vitali's theorem without uniform boundedness. Publ. Mat. 60 (2016), no. 2, 311--334. doi:10.5565/PUBLMAT_60216_03.

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