Domains of holomorphy Nestoridis, V.
Annales mathématiques du Québec,
04/2018, Volume:
42, Issue:
1
Journal Article
Peer reviewed
We give a simple proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire’s Category Theorey and Montel’s Theorem. We ...also obtain generalizations by demanding that the non-extentable functions belong to a particular class of functions
X
=
X
(
Ω
)
⊂
H
(
Ω
)
. We show that the set of non-extendable functions not only contains a
G
δ
-dense subset of
X
(
Ω
)
, but it is itself a
G
δ
-dense set. We give an example of a domain in
C
which is a
H
(
Ω
)
-domain of holomorphy but not a
A
(
Ω
)
-domain of holomorphy.
In this paper I present my first proof regarding the existence of universal Taylor series on the disc where the universal approximation was required on the boundary as well. It is a modification of a ...construction giving a negative answer to a question of S. Pichorides, where the approximation was valid only on the boundary of the disc. There was no use of Baire’s theorem in the above proofs. J.-P. Kahane suggested to use Baire’s theorem which yields stronger results with simpler proofs. Later, Baire’s theorem was systematically used in order to establish new generic universalities.
Given a pair of topological vector spaces
X
,
Y
where
X
is a proper linear subspace of
Y
it is examined whether
Y
\
X
is residual in
Y
(topological genericity), whether
Y
\
X
contains a dense linear ...subspace of
Y
except 0 (algebraic genericity) and whether
Y
\
X
contains a closed infinite dimensional subspace of
Y
except 0 (spaceability). In the present paper the spaces
X
and
Y
are either sequence spaces or spaces of analytic functions on the unit disc regarded as sequence spaces via the identification of a function with the sequence of its Taylor coefficients. For the spaces under consideration we give an affirmative answer to each of these questions providing general proofs which extend previous results.
We establish generic existence of Universal Taylor Series on products
Ω
=
∏
Ω
i
of planar simply connected domains
Ω
i
where the universal approximation holds on products
K
of planar compact sets ...with connected complements provided
K
∩
Ω
=
∅
. These classes are with respect to one or several centers of expansion and the universal approximation is at the level of functions or at the level of all derivatives. Also, the universal functions can be smooth up to the boundary, provided that
K
∩
Ω
¯
=
∅
and
{
∞
}
∪
C
\
Ω
¯
i
is connected for all
i
. All previous kinds of universal series may depend on some parameters; then the approximable functions may depend on the same parameters, as it is shown in the present paper.
Using a recent Mergelyan type theorem for products of planar compact sets, we establish generic existence of universal Taylor series on products of planar simply connected domains
Ω
i
,
i
=
1
,
…
,
d
.... The universal approximation is realized by partial sums of the Taylor development of the universal function on products of planar compact sets
K
i
,
i
=
1
,
…
,
d
such that
C
-
K
i
is connected and for at least one
i
0
the set
K
i
0
is disjoint from
Ω
i
0
.
Using complex methods combined with Baire’s Theorem, we show that one-sided extendability, extendability, and real analyticity are rare phenomena on various spaces of functions in the topological ...sense. These considerations led us to introduce the
p
-continuous analytic capacity and variants of it,
p
∈
{
0
,
1
,
2
,
…
}
∪
{
∞
}
, for compact or closed sets in
C
. We use these capacities in order to characterize the removability of singularities of functions in the spaces
A
p
.
Concurrent universal Padé approximation Makridis, K.; Nestoridis, V.
Journal of mathematical analysis and applications,
05/2021, Volume:
497, Issue:
2
Journal Article
Peer reviewed
We prove concurrent universal Padé approximation for several universal Padé approximants of several types. Our results are generic in the space of holomorphic functions, in the space of formal power ...series as well as in a subspace of A∞. These results are valid for one center of expansion or for several centers as well. The Padé approximants allow generic approximation on arbitrary compact sets, not necessarily having connected component, in contrast with the partial sums of power series. We also establish affine genericity for a class of universal functions.
We show that the set of frequently universal harmonic functions on a tree T contains a vector space except 0 which is dense in the space of harmonic functions on T seen as subset of CT. In order to ...prove this we replace the complex plane C by any separable Fréchet space E and we repeat all the theory.
We give an abstract framework for the theory of universal series, from which we deduce easily and in a unified way most of the existing results as well as new and stronger statements.
Recently, harmonic functions and frequently universal harmonic functions on a tree T have been studied, taking values on a separable Fréchet space E over the field C or R. In the present paper, we ...allow the functions to take values in a vector space E over a rather general field F. The metric of the separable topological vector space E is translation invariant and instead of harmonic functions we can also study more general functions defined by linear combinations with coefficients in F. We don't assume that E is complete and therefore we present an argument avoiding Baire's theorem.
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