Lectures on Complex Approximation

I: Approximation by Series Expansions and by Interpolation.- I. Representation of complex functions by orthogonal series and Faber series.- §1. The Hilbert space L2(G).- A. Definition of L2(G).- B. L2(G) as a Hilbert space.- §2. Orthonormal systems of polynomials in L2(G).- A. Construction of ON systems; Gramian matrix.- A1. The Gram-Schmidt orthogonalization process.- A2. The Gramian matrix.- A3. A special case: Polynomials in L2(G).- B. Zeros of orthogonal polynomials.- C. Asymptotic representation of the ON polynomials.- Remark about §2.- §3. Completeness of the polynomials in L2(G).- A. The problem and examples.- B. Domains with the PA property.- C. Domains not having the PA property.- C1. Slit domains.- C2. Moon-shaped domains.- Remarks about §3.- §4. Expansion with respect to ON systems in L2(G).- A. ON expansions in Hilbert space.- B. ON expansions in the space L2(G).- C. The quality of the approximation if f is analytic in $$ \bar G $$.- Remarks about §4.- §5. The Bergman kernel function.- A. Introduction and properties of the kernel function.- B. Series representation of the Bergman kernel function.- C. Construction of conformal mappings with the Bergman kernel function.- C1. The connection between K and conformal mapping.- C2. The Bieberbach polynomials.- C3. The use of singular functions in the ON process.- D. Additional applications of the Bergman kernel function.- D1. Domains with the mean-value property.- D2. Representation of $$ \int_{ - 1}^{ + 1} {f(x)dx} $$ as an area integral.- Remark about §5.- §6. The quality of the approximation; Faber expansions.- A. Boundary behavior of Cauchy integrals.- B. Faber polynomials and Faber expansions.- C. The Faber mapping as a bounded operator.- C1. Curves of bounded rotation.- C2. The Faber mapping T.- D. The quality of approximation inside a curve of bounded rotation.- D1 Preparations; uniform convergence.- D2. The modulus of continuity of the Cauchy integral corresponding to h.- D3. The quality of the approximation.- E. Report on additional results.- E1 Additional uniform estimates.- E2. Local estimates.- Remarks about §6.- II. Approximation by interpolation.- §1. Hermite’s interpolation formula.- A. The interpolating polynomial.- B. Special cases of Hermite’s formula.- §2. Interpolation in uniformly distributed points; Fejér points, Fekete points.- A. Preparations; rough statement about convergence.- B. General convergence theorem of Kalmár and Walsh.- C. The system of Fejér points.- D. The system of Fekete points.- Remarks about §2.- §3. Approximation on more general compact sets; Runge’s theorem.- A. Again: Interpolation in Fekete points.- B. Runge’s approximation theorem.- Remark about §3.- §4. Interpolation in the unit disk.- A. Interpolation on {z: | z | = r}, r < 1.- B. Interpolation on {z: | z | = 1}.- C. Approximation by rational functions.- Remarks about §4.- II: General Approximation Theorems in the Complex Plane.- III. Approximation on compact sets.- §1. Runge’s approximation theorem.- A. General Cauchy formula.- B. Runge’s theorem.- C. The pole shifting method.- §2. Mergelyan’s theorem.- A. Formulation of the result; special cases; consequences.- B. Preparations for the proof.- B1. Tietze’s extension theorem.- B2. A representation formula.- B3. Koebe’s ¼-Theorem.- B4. Mergelyan’s lemma.- C. Proof of Mergelyan’s theorem.- Remark about §2.- §3. Approximation by rational functions.- A. Swiss cheese.- A1. Alice Roth’s construction.- A2. Swiss cheese with interior points.- A3. Swiss cheese with two components.- A4. Accumulation of holes at the diameter of $$ \mathbb{D} $$.- B. Preparations for Bishop’s theorem.- B1. An integral transform.- B2. Partition of unity.- C. Bishop’s localization theorem and applications.- C1. The localization theorem.- C2. Applications of Bishop’s theorem.- D. Vitushkin’s theorem; a report.- Remarks about §3.- §4. Roth’sfusion lemma.- A. The fusion lemma.- B. A new proof of Bishop’s theorem.- Remark about §4.- IV. Approximation on closed sets.- §1. Uniform approximation by meromorphic functions.- A. Statement of the problem.- B. Roth’s approximation theorem.- C. Special cases of the approximation theorem.- C1. The one-point compactification G* of G; connectedness of G*\F.- C2. Three sufficient criteria for meromorphic approximation.- D. Characterization of the sets, where meromorphic approximation is possible.- §2. Uniform approximation by analytic functions.- A. Moving the poles of meromorphic functions.- B. Preliminary topological remarks.- C. Arakeljaii’s approximation theorem.- C1. Approximation of meromorphic functions by analytic functions.- C2. Arakeljan’s theorem.- Remarks about §2.- §3. Approximation with given error functions.- A. The problem; Carleman’s theorem.- A1. Tangential approximation; ?-approximation.- A2. Two lemmas.- A3. Carleman’s theorem.- B. The special case where F is nowhere dense.- B1. Sufficient conditions for ?-approximation.- B2. Tangential approximation if F° = ?.- C. Nersesjan’s theorem.- C1. Condition (A); a lemma.- C2. Nersesjan’s theorem.- Remarks about §3.- §4. Approximation with certain error functions.- A. ?-approximation without condition (A).- B. Growth of the approximating function.- C. The special case F = ?.- §5. Some applications of the approximation theorems.- A. Radial boundary values of entire functions.- B. Boundary behavior of functions analytic in the unit desk.- B1. A general approximation theorem.- B2. The Dirichlet problem for radial limits.- C. Approximation and uniqueness theorems.- D. Various further constructions.- D1. Prescribed boundary behavior along countably many curves.- D2. Analytic functions with prescribed cluster sets.- D3. Schneider’s noodles.- D4. Julia directions of entire functions.- Remarks about §5.- References.