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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.

Inhaltsverzeichnis

Frontmatter

Approximation by Series Expansions and by Interpolation

Frontmatter

Chapter I. Representation of Complex Functions by Orthogonal Series and Faber Series

Abstract
As is well known, one of the most important methods of representing functions defined on real or complex domains with the help of simpler functions is the method of series expansions. The theory of convergence for functions defined on complex domains, especially for analytic functions, is considerably simpler than for functions defined on real domains. Since we are generally interested in analytic functions, we shall mainly be concerned with series developments in the space L 2(G). The first four sections of this chapter are devoted to this topic. An important element in the space L 2(G) is the Bergman kernel function, which is useful for the construction of conformal mappings. We talk about the Bergman kernel function in Section 5. Finally, in Section 6, we present the expansion of functions in Faber polynomials in order to obtain certain theorems on the quality of approximation by polynomials.
Dieter Gaier

Chapter II. Approximation by Interpolation

Abstract
Next to series expansion, interpolation represents a further important tool for the approximation of functions; we now turn to this method.
Dieter Gaier

General Approximation Theorems in the Complex Plane

Frontmatter

Chapter III. Approximation on Compact Sets

Abstract
For the approximation of a function f on a compact set K polynomials or, more generally, rational functions can be utilized. The situation is simple if f is analytic on K, whereas a weakening of this assumption requires a much greater effort. We begin with Runge’s theorem; a weak version of it has already been proved, but the following proof will make Chapter III independent of the preceding chapters.
Dieter Gaier

Chapter IV. Approximation on Closed Sets

Abstract
So far, we have approximated functions defined on a compact set K ⊂ ℂ, and the approximating functions were polynomials or rational functions. Now we shall approximate functions defined on a set F that is closed in a domain G. Functions analytic or meromorphic in G will serve as approximating functions. In the special case where G = ℂ, one obtains approximation by entire functions. Here the rate of approximation (as z → ∞) also plays a role. Several of these theorems can be used to construct analytic functions with complicated boundary behavior; we deal with these questions at the end of the chapter, in §5.
Dieter Gaier

Backmatter

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