Lectures on Complex Approximation

The theory of General Relativity, after its invention by Albert Einstein, remained for many years a monument of mathemati­ cal speculation, striking in its ambition and its formal beauty, but quite separated from the main stream of modern Physics, which had centered, after the early twenties, on quantum mechanics and its applications. In the last ten or fifteen years, however, the situation has changed radically. First, a great deal of significant exper~en­ tal data became available. Then important contributions were made to the incorporation of general relativity into the framework of quantum theory. Finally, in the last three years, exciting devel­ opments took place which have placed general relativity, and all the concepts behind it, at the center of our understanding of par­ ticle physics and quantum field theory. Firstly, this is due to the fact that general relativity is really the "original non-abe­ lian gauge theory," and that our description of quantum field in­ teractions makes extensive use of the concept of gauge invariance. Secondly, the ideas of supersymmetry have enabled theoreticians to combine gravity with other elementary particle interactions, and to construct what is perhaps the first approach to a more finite quantum theory of gravitation, which is known as super­ gravity.

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.