Lectures on Complex Approximation -

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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 dollars dollars \bar G dollars dollars.- 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 dollars dollars \int_{ - 1}^{ + 1} {f(x)dx} dollars dollars 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; Fej?r points, Fekete points.- A. Preparations; rough statement about convergence.- B. General convergence theorem of Kalm?r and Walsh.- C. The system of Fej?r 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: