Publicacions Matemàtiques

Vitali's theorem without uniform boundedness

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

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Abstract

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

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

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

Permanent link to this document
http://projecteuclid.org/euclid.pm/1468242038

Digital Object Identifier
doi:10.5565/PUBLMAT_60216_03

Mathematical Reviews number (MathSciNet)
MR3521494

Zentralblatt MATH identifier
1347.41002

Subjects
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]

Keywords
Rapid convergence convergence in capacity pluripolar set relative capacity

Citation

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. http://projecteuclid.org/euclid.pm/1468242038.


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