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We show that a -smooth mapping on an open subset of , , can be approximated in a fine topology and together with its derivatives by a restriction of a holomorphic mapping with explicitly described domain. As a corollary we obtain a generalisation of the Carleman-Scheinberg theorem on approximation by entire functions.
Holomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions...
The -weighted Besov spaces of holomorphic functions on the unit ball in are introduced as follows. Given a function of regular variation and , a function holomorphic in is said to belong to the Besov space if
where is the volume measure on and stands for the fractional derivative of . The holomorphic Besov space is described in the terms of the corresponding space. Some projection theorems and theorems on existence of the inversions of these projections are proved. Also,...
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