This paper is devoted to a study of harmonic mappings of a harmonic space on a harmonic space which are related to a family of harmonic mappings of into . In this way balayage in may be reduced to balayage in . In particular, a subset of is polar if and only if is polar. Similar result for thinness. These considerations are applied to the heat equation and the Laplace equation.
We construct Almansi decompositions for a class of differential operators, which include powers of the classical Laplace operator, heat operator, and wave operator. The decomposition is given in a constructive way.
We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a δ-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on δ. We apply the obtained estimates...
The survey collects many recent advances on area Nevanlinna type classes and related spaces of analytic functions in the unit disk concerning zero sets and factorization representations of these classes and discusses approaches, used in proofs of these results.