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The harmonic Faber operator
(2018)

P. K. Suetin points out in the beginning of his monograph "Faber Polynomials and Faber Series" that Faber polynomials play an important role in modern approximation theory of a complex variable as they are used in representing analytic functions in simply connected domains, and many theorems on approximation of analytic functions are proved with their help [50]. In 1903, the Faber polynomials were firstly discovered by G. Faber. It was Faber's aim to find a generalisation of Taylor series of holomorphic functions in the open unit disc D in the following way. As any holomorphic function in D has a Taylor series representation f(z)=\sum_{\nu=0}^{\infty}a_{\nu}z^{\nu} (z\in\D) converging locally uniformly inside D, for a simply connected domain G, Faber wanted to determine a system of polynomials (Q_n) such that each function f being holomorphic in G can be expanded into a series
f=\sum_{\nu=0}^{\infty}b_{\nu}Q_{\nu} converging locally uniformly inside G. Having this goal in mind, Faber considered simply connected domains bounded by an analytic Jordan curve. He constructed a system of polynomials (F_n) with this property. These polynomials F_n were named after him as Faber polynomials. In the preface of [50], a detailed summary of results concerning Faber polynomials and results obtained by the aid of them is given. An important application of Faber polynomials is e.g. the transfer of known assertions concerning polynomial approximation of functions belonging to the disc algebra to results of the approximation of functions being continuous on a compact continuum K which contains at least two points and has a connected complement and being holomorphic in the interior of K. In this field, the Faber operator denoted by T turns out to be a powerful tool (for an introduction, see e.g. D. Gaier's monograph). It
assigns a polynomial of degree at most n given in the monomial basis \sum_{\nu=0}^{n}a_{\nu}z^{\nu} with a polynomial of degree at most n given in the basis of Faber polynomials \sum_{\nu=0}^{n}a_{\nu}F_{\nu}. If the Faber operator is continuous with respect to the uniform norms, it has a unique continuous extension to an operator mapping the disc algebra onto the space of functions being continuous on the whole compact continuum and holomorphic in its interior. For all f being element of the disc algebra and all polynomials P, via the obvious estimate for the uniform norms ||T(f)-T(P)||<= ||T|| ||f-P||, it can be seen that the original task of approximating F=T(f) by polynomials is reduced to the polynomial approximation of the function f. Therefore, the question arises under which conditions the Faber operator is continuous and surjective. A fundamental result in this regard was established by J. M. Anderson and J. Clunie who showed that if the compact continuum is bounded by a rectifiable Jordan curve with bounded boundary rotation and free from cusps, then the Faber operator with respect to the uniform norms is a topological isomorphism. Now, let f be a harmonic function in D. Similar as above, we find that f has a uniquely determined representation f=\sum_{\nu=-\infty}^{\infty}a_{\nu}p_{\nu}
converging locally uniformly inside D where p_{n}(z)=z^{n} for n\in\N_{0} and p_{-n}(z)=\overline{z}^{n} for n\in\N}. One may ask whether there is an analogue for harmonic functions on simply connected domains G. Indeed, for a domain G bounded by an analytic Jordan curve, the conjecture that each function f being harmonic in G has a uniquely determined representation f=\sum_{\nu= \infty}^{\infty}b_{\nu}F_{\nu} where F_{-n}(z)=\overline{F_{n}(z\)} for n\inN, converging locally uniformly inside G, holds true. Let now K be a compact continuum containing at least two points and having a connected complement. A main component of this thesis will be the examination of the harmonic Faber operator mapping a harmonic polynomial given in the basis of the harmonic monomials \sum_{\nu=-n}^{n}a_{\nu}p_{\nu} to a harmonic polynomial given as \sum_{\nu=-n}^{n}a_{\nu}F_{\nu}.
If this operator, which is based on an idea of J. Müller, is continuous with respect to the uniform norms, it has a unique continuous extension to an operator mapping the functions being continuous on \partial\D onto the continuous functions on K being
harmonic in the interior of K. Harmonic Faber polynomials and the harmonic Faber operator will be the objects accompanying us throughout
our whole discussion. After having given an overview about notations and certain tools we will use in our consideration in the first chapter, we begin our studies with an introduction to the Faber operator and the harmonic Faber operator. We start modestly and consider domains bounded by an analytic Jordan curve. In Section 2, as a first result, we will show that, for such a domain G, the harmonic Faber operator has a unique continuous extension to an operator mapping the space of the harmonic functions in D onto the space
of the harmonic functions in G, and moreover, the harmonic Faber
operator is an isomorphism with respect to the topologies of locally
uniform convergence. In the further sections of this chapter, we illumine the behaviour of the (harmonic) Faber operator on certain function spaces. In the third chapter, we leave the situation of compact continua bounded by an analytic Jordan curve. Instead we consider closures of domains bounded by Jordan curves having a Dini continuous curvature. With the aid of the concept of compact operators and the Fredholm alternative, we are able to show that the harmonic Faber operator is a topological isomorphism. Since, in particular, the main result of the third chapter holds true for closures K of domains bounded by analytic Jordan curves, we can make use of it to obtain new results concerning the approximation of functions being continuous on K and harmonic in the interior of K by harmonic polynomials. To do so, we develop techniques applied by L. Frerick and J. Müller in [11] and adjust them to our setting. So, we can transfer results about the classic Faber operator to the harmonic Faber operator. In the last chapter, we will use the theory of harmonic Faber polynomials
to solve certain Dirichlet problems in the complex plane. We pursue
two different approaches: First, with a similar philosophy as in [50],
we develop a procedure to compute the coefficients of a series \sum_{\nu=-\infty}^{\infty}c_{\nu}F_{\nu} converging uniformly to the solution of a given Dirichlet problem. Later, we will point out how semi-infinite programming with harmonic Faber polynomials as ansatz functions can be used to get an approximate solution of a given Dirichlet problem. We cover both approaches first from a theoretical point of view before we have a focus on the numerical implementation of concrete examples. As application of the numerical computations, we considerably obtain visualisations of the concerned Dirichlet problems rounding out our discussion about the harmonic Faber polynomials and the harmonic Faber operator.