Convergence of Sequences of Rational Functions on ℂ n

• Published in: Vietnam journal of mathematics Vol. 45; no. 4; pp. 669 - 679
• Main Authors: Dau Hoang Hung, Le Thanh Hung
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.12.2017
• Subjects: Continuity (mathematics); Convergence; Rational functions
• ISSN: 2305-221X; 2305-2228
• DOI: 10.1007/s10013-017-0246-y

Let A : = { χ m } m ≥ 1 be a sequence of continuous, real valued functions defined on [ 0 , ∞ ) . We say that a sequence {fm}m ≥ 1 of measurable functions defined on X ⊂ D ⊂ ℂ n is convergent in capacity (relative to D) with respect to the weight sequence A to a function f if χm(|f − fm|2) converges to 0 in capacity on X. We are interested in finding conditions (on A ) so that every sequence {rm}m ≥ 1 of rational functions on ℂ n converges in capacity with respect to A to a holomorphic function f defined on a bounded domain D ⊂ ℂ n provided that the convergence holds only pointwise on a small subset of D.