Uber die analytische Kapazität
01 Dec 1960-Archiv der Mathematik (Springer Science and Business Media LLC)-Vol. 11, Iss: 1, pp 270-277
About: This article is published in Archiv der Mathematik.The article was published on 1960-12-01. It has received 32 citations till now.
Citations
More filters
TL;DR: In this article, the authors define the analytic capacity of sets and define the connection between the capacity of a set and measures, and give a generalization of the capacity analogue to the theorem on density points.
Abstract: CONTENTSIntroductionChapter I. The analytic capacity of sets § 1. Definition and some properties of analytic capacity § 2. The connection between the capacity of a set and measures § 3. On removable singularities of analytic functions § 4. The analytic C-capacity of sets § 5. Estimates of the coefficients in the Laurent series § 6. The change in the capacity under a conformal transformation of a set Chapter II. The separation of singularities of functions § 1. The construction of a special system of partitions of unity § 2. Integral representations of continuous functions § 3. Separation of singularities § 4. The approximation of functions in parts § 5. Approximation of functions on sets with empty inner boundary § 6. The additivity of capacity for some special partitions of a setChapter III. Estimation of the Cauchy integral § 1. Statement of the result § 2. Estimate of the Cauchy integral § 3. Estimation of the Cauchy integral along a smooth Lyapunov curve § 4. Proof of the principal theorem § 5. Some consequences § 6. A refinement of the Maximum Principle and the capacity analogue to the theorem on density pointsChapter IV. Classification of functions admitting an approximation by rational fractions § 1. Examples of functions that cannot be approximated by rational fractions § 2. A criterion for the approximability of a function § 3. Properties of the second coefficient in the Laurent series § 4. Proof of the principal lemma § 5. Proof of the theorems of § 2Chapter V. The approximation problem for classes of functions § 1. Removal of the poles of approximating functions from the domain of analyticity of the function being approximated § 2. Necessary conditions for the algebras to coincide § 3. A criterion for the equality of the algebras § 4. Geometrical examples § 5. Some problems in the theory of approximationChapter VI. The approximation of functions on nowhere dense sets § 1. The instability of capacity § 2. A capacity criterion for the approximability of functions on nowhere dense sets § 3. Theorems on the approximation of continuous functions in terms of Banach algebras § 4. A capacity characterization of the Mergelyan function and of peak pointsReferences
228 citations
Book•
09 Dec 2015
TL;DR: The Cauchy integral theorem and its consequences are discussed in this paper, with a focus on spaces of analytic functions, fractional linear transformations, conformal maps, and elliptic functions.
Abstract: * Preliminaries* The Cauchy integral theorem: Basics Consequences of the Cauchy integral formula* Chains and the ultimate Cauchy integral theorem* More consequences of the CIT* Spaces of analytic functions* Fractional linear transformations* Conformal maps* Zeros of analytic functions and product formulae* Elliptic functions* Selected additional topics* Bibliography* Symbol index* Subject index* Author index* Index of capsule biographies
101 citations
01 Jan 1994
TL;DR: In this article, the problem of qualitative approximation by holomorphic functions of one complex variable belonging to some fixed class is studied. But the problem is not restricted to the class under consideration and its associated capacity.
Abstract: In this paper we are primarily interested in problems of qualitative approximation by holomorphic functions of one complex variable belonging to some fixed class, that is defined by restricting the growth of the functions (L p , 1 < p ≤ ∞) or by requiring certain smoothness (Lip s or C m ). Part of the approximation problem consists in understanding the removable sets for the class under consideration and its associated capacity.
41 citations
Posted Content•
TL;DR: In this article, a survey on the Ahlfors function and the weaker circle maps is presented, i.e. those (branched) maps effecting the conformal representation upon the disc of a compact bordered Riemann surface.
Abstract: This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.
41 citations
Posted Content•
TL;DR: In this paper, Riemann, Ahlfors, Rohlin, and Rohlin discuss the current consensus about Hilbert's 16th in degree 8, a nearly finished piece of mathematics, thanks to heroic breakthroughs by Viro, Fiedler, Korchagin, Shustin, Chevallier, Orevkov, yet still leaving undecided six tantalizing bosons among a menagerie of 104 logically possible distributions of ovals.
Abstract: This text is intended to become in the long run Chapter 3 of our long saga dedicated to Riemann, Ahlfors and Rohlin. Yet, as its contents evolved as mostly independent (due to our inaptitude to interconnect both trends as strongly as we wished), it seemed preferable to publish it separately. More factually, our account is an attempt to get familiarized with the current consensus about Hilbert's 16th in degree 8. This is a nearly finished piece of mathematics, thanks heroic breakthroughs by Viro, Fiedler, Korchagin, Shustin, Chevallier, Orevkov, yet still leaving undecided six tantalizing bosons among a menagerie of 104 logically possible distributions of ovals (respecting B\'ezout, Gudkov periodicity, and the Fiedler-Viro imparity law sieving away 4+36 schemes). This quest inevitably involves glimpsing deep into the nebula referrable to as Viro's patchworking, and the likewise spectacular obstructional laws of Fiedler, Viro, Shustin, Orevkov. In the overall, the game is much comparable to a pigeons hunting video-game, where 144 birds are liberated in nature, with some of them strong enough to fly higher and higher in the blue sky as to rejoin the stratospheric paradise of eternal life (construction of a scheme in the algebro-geometric category). Some other, less fortunate, birds were killed (a long time ago) and crashed down miserably over terrestrial crust (prohibition). Alas, the hunt is unfinished with still six birds, apparently too feeble to rally safely the paradise, yet too acrobatic for any homosapiens being skillful enough to shoot them down.
40 citations
References
More filters
TL;DR: The most useful conformal invar iants are obtained by solving conformMly invar ian t ex t remal problems as discussed by the authors, and their usefulness derives from the fact that they must automatically satisfy a principle of majorizat ion.
Abstract: The most useful conformal invar iants are obtained by solving conformMly invar ian t ex t remal problems. The i r usefulness derives f rom the fac t tha t they must automat ical ly satisfy a principle of majorizat ion. There is a r ich variety of such problems, and if we would aim at completeness this paper would assume forbidding proport ions. We shall therefore l imit ourselves to a few part icular ly simple i n v a r i a n t s and study thei r propert ies and in ter re la t ions in considerable detail. Each class of invar ian ts is connected wi th a category of null-sets, which by this very fac t en ter na tura l ly in funct iontheoret ic considerat ions. A null-set is the complement of a region for which a cer ta in conformal invar iant degenerates. Inequal i t ies between invar iants lead to inclusion relat ions between the corresponding classes of null-sets. Th roughou t this paper Y2 will denote an open region in the extended z plane, and Zo will be a dis t inguished point in t~. Most results will be formula ted for the case z 0 ~ c~, but the t rans i t ion to z o = c~ is always trivial. In some instances the la t te r case offers formal advantages. We shall consider classes of funct ions f(z) which are analyt ic and singlevalued in some region t). F o r a general class ~ the region t~ is al lowed to vary with f , but the subclass of funct ions in a fixed region t~ will be denoted by ~(t2). For ZoE ~ we in t roduce the quant i ty
419 citations
182 citations
TL;DR: In this paper, the fundamental distortion theorems of Schlicht conformal mapping theory have been developed by Grunsky by a method which is able to apply here to obtain the fundamental distortions for bounded functions.
Abstract: Introduction. This paper is concerned with extremal problems in the family of bounded analytic functions in a multiply-connected domain D, and it is concerned with extremal problems for the mean modulus f cIf Ids of meromorphic functions f in D taken over the boundary C of D. These two types of problems are shown to be closely related, and solutions are obtained simultaneously for both types by a method of contour integration. An altogether analogous method was exploited by Grunsky in his thesis [8](2) for the investigation of schlicht functions in a multiply-connected domain. Thus it is interesting to remark that the fundamental distortion theorems of schlicht conformal mapping theory have been developed by Grunsky by a method which we are able to apply here to obtain the fundamental distortion theorems for bounded functions. It will appear, then, that the generalization of Schwarz's lemma to multiply-connected domains and the generalization of the Koebe distortion theorem can be carried out by a unified technique(3). We shall find in addition to this that, while the recent papers of Bergman and Schiffer [5, 16] have developed a close relationship between the theory of schlicht canonical mappings and the Bergman kernel function [3], we are able to develop here a relationship between the theory of bounded functions and the Szego kernel function [19]. Thus the Szego kernel function does for the theory of distortion of bounded functions what the Bergman kernel function does for the theory of distortion of schlicht functions. We point out that both these kernel functions are actually differentials, and that in the Szego case one is dealing with length and in the Bergman case one is dealing with area. Thus the mean modulus fc f I ds, or ffD IfI 2dxdy, should be thought of as a length, or area, and not as a mean modulus. All these remarkable relationships are brought to light by using the simple boundary relations satisfied by the classical domain functions, such as
85 citations
47 citations
5 citations