Let \(\Omega \) be a Stein manifold. If \(D\) is an open subset of \(\Omega \), we
denote by \(A\left( D\right) \) the locally convex space of all functions
analytic in \(D\) with the topology of locally uniform convergence in \(D\). For
an arbitrary set \(E\subset \Omega ,\) \(A\left( E\right) \) is a set of all
analytic germs on \(E\) considered with the locally convex topology of the
inductive limit:
\[
A\left( E\right) = \textrm{limind}_{G\in \mathcal{O}\left( E\right)
}A\left( G\right) ,
\]
where \(\mathcal{O}\left( E\right) \) is the set of all open neighborhoods of \(
E\).
Our goal is to give a survey of results on Schauder bases in those spaces
(especially, for the cases when \(E\) is an open or compact set in \(\Omega \)
). Main attention will be paid to the following topics: existence of bases
in spaces \(A\left( E\right) \), their construction and structure, extendible
bases; orthogonal, doubly orthogonal and interpolation bases, application
to isomorphic classification of spaces of analytic functions; applications
to approximation, interpolation and extension of analytic functions. The
one-dimensional case is considered separately, since some results can be
proved easier than their multivariate counterparts, the
others have no many-dimensional analogs at all. This is due to some
specific one-dimensional tools (like the fundamental algebra theorem, or
the Grothendieck-Köthe-Silva duality, or the potential theory in \(
\mathbb{R}^{2}=\mathbb{C}\)). For \(\dim \Omega \geq 2\) the pluripotential
theory and functional analysis methods play main role. Some long standing
open problems will be discussed.
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