Vitali's theorem without uniform boundedness

Abstract

Let {fm} m≥1 be a sequence of holomorphic functions defined on a bounded domain D ⊂ C n or a sequence of rational functions (1 ≤ deg rm ≤ m) defined on C n . We are interested in finding sufficient conditions to ensure the convergence of {fm} m≥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.

Convergence With Respect to Admissible Sequences of Weights for a Holomorphic Space of a Sequence of Hilbert-valued Rational Functions

“…Thus {v m } m≥1 does not converge to −∞ uniformly on X ′ . Finally, by Proposition 5.1, applying Lemma 2.9 in [3] to the sequences {u m } m≥1 and {v m } m≥1 we get {w m (∥(r m − f )(•)∥ 2 } m≥1 converges to 0 in capacity on D. This means that (a) is proved.…”

“…(b) By Proposition 5.1 the sequence {ψ m } m≥1 convergences to −∞ uniformly on compact subsets on D. We also note that by (5.2) and the condition (A 3 (ii)) again, the sequence {v m } m≥1 is uniformly bounded from above on compact sets of C N . Since {v m } m≥1 does not converge to −∞ uniformly on X ′ (see the proof of (a)), using again Lemma 3.8 in [3] we deduce that lim sup m→∞ v m > −∞ on D \ E with…”

“…Assume the contrary, we can assume that the sequence ψ m itself converges almost everywhere on D to ψ. Applying Lemma 3.8 in [3] we have lim sup m→∞ ψ m = ψ outside a pluripolar set. Note that, it follows from (5.2) and the condition (A 3 (ii)) that…”

On the rapidly convergence in capacity of the sequence of holomorphic functions

“…We need the following auxiliary fact giving a sufficient condition for a sequence of measurable functions converging in capacity to 0. We would like to repeat a reasoning due to [3] for the reader convenience. For N ≥ 1 we let…”

“…Throughout this paper, we always assume that 1 ≤ deg r m ≤ m i.e., the numerator and the denominator of r m are non-constant polynomials of degree at most m. Let X be a Borel subset which lies either in D or ∂D. In this paper, we will refine the techniques in [3] concerning with sufficient conditions that guarantees convergence in some sense of { f m − r m } m≥1 to 0 on D as soon as the convergence occurs pointwise on X. By a classical theorem of Vitali, under the additional conditions that {r m } m≥1 and { f m } m≥1 are uniformly bounded on compact sets of D and X is not contained in any proper analytic subset of D, the sequence { f m − r m } m≥1 converges uniformly to 0 on compact sets of D provided that the convergence holds pointwise only on X.…”