Zur Theorie der nichtlinearen Tschebyscheff-Approximation mit Nebenbedingungen

This paper deals with questions of nonlinear Tschebyscheff-approximation theory, the approximations being constrained by nonlinear relations. We assume the approximating functions depending Fréchet-differentiable on a parameter and the constraints satisfying certain regularity and differentiability properties. Under these hypotheses in the main theorem we give necessary conditions to characterisize best approximations. Using these results, some problems in approximating functions, the best approximations being regarded to satisfy interpolatory conditions, are discussed. We deduce, that in this case best approximations admit a characterisation by generalized alternants.

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Erster Teil einer gekürzten Fassung der Dissertation des Verfassers [1968].     

APPROXIMATION BY VECTOR-VALUED FUNCTIONS

ABSTRACT

The theory of H-sets was first propounded by L. Collatz [3] and [4]. This concept has been shown to be useful in the study of uniform approximation, and we here consider the form H-sets take in this setting of vector-valued functions and prove a general characterization theorem. A similar exposition for the linear real-valued case can be found in [1].     

über Bewegungen in der absoluten Geometrie

In this paper we improve a theorem on the representation of isometries in (generalized) euclidean and non-euclidean geometries of arbitrary dimension presented in a previous paper [3]. As an application we give a characterization of isometries in spaces of three dimensions.

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W. BURAU zum 70. Geburtstag gewidmet     

Gleichverteilte Folgen in lokalkompakten Gruppen

In this paper we consider concepts of uniform distribution in locally compact groups in connection with representation theory. We study questions concerning the existence and quality of uniformly distributed sequences in connection with the structure of general locally compact groups (including also several examples of particular groups), certain separability-conditions, and the structure of the according representation space (the latter primarily for the circle-group). For connections to known concepts the reader is referred to [29], [14], [25], [48], [21], [53]; [19]; [26], [17]; [9]; [3]; [40].

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Herrn Prof. Dr. E. Hlawka zum 60. Geburtstag gewidmet     

A multiplicity result for a class of nonlinear boundary value problems of the ambrosetti-prodi type

Summary In this paper we prove the existence of at least two solutions for a nonlinear elliptic boundary value problem of the Ambrosetti-Prodi type in the case when there exist two suitable supersolutions. This result follows from an abstract theorem of existence of multiple solutions which is a variant of a result due to Ambrosetti and Hess, [5].

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Riassunto In questo lavoro si prova che per un problema al contorno del tipo Ambrosetti-Prodi esistono almeno due soluzioni se esistono due opportune soprasoluzioni. Tale risultato discende da un teorema astratto di esistenza di soluzioni multiple che é una variante di un risultato di Ambrosetti ed Hess, [5].

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Paper written under the auspices of G.N.A.F.A., C.N.R., Italy.     

A Study in the BV Space of a Denoising—Deblurring Variational Problem

In this paper we study, in the framework of functions of bounded variation, a general variational problem arising in image recovery, introduced in [3]. We prove the existence and the uniqueness of a solution using lower semicontinuity results for convex functionals of measures. We also give a new and fine characterization of the subdifferential of the functional, together with optimality conditions on the solution, using duality techniques of Temam for the theory of time-dependent minimal surfaces. We study the associated evolution equation in the context of nonlinear semigroup theory and we give an approximation result in continuous variables, using Γ -convergence. Finally, we discretize the problems by finite differences schemes and we present several numerical results for signal and image reconstruction. Accepted 5 March 2001. Online publication 20 June 2001.     

Eine gruppentheoretische Kennzeichnung ebener kommutativer punktiert-affiner Inzidenzgruppen

In this paper we give a group-theoretical characterization of plane commutative punctured affine incidence groups as introduced by G.P. Kist in [1].

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag     

Über Die Konvergenz von Einzelschrittverfahren zur Minimierung konvexer Funktionen

A first convergence proof for free steering minimization methods was given by SCHECHTER [3]. The function f to be minimized was assumed by him to be twice continously differentiable and strictly convex. These conditions were weakened by ELKIN [1], who considered continously differentiable and uniformly convex functions and who employed a slight modification of SCHECHTER's convergence proof. The following article gives a new proof for convergence of free steering methods. We assume f to be strictly convex and continously differentiable.

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Aus Kap. I 1. der Dissertation des Verfassers [2].     

Ein Konstruktionsverfahren bei allgemeiner linearer Approximation

In order to solve the linear approximation problem the Remes-algorithm is quite suitable in case of a Haar subspace. Without this assumption one must modify the algorithm even to ensure the practicability (cf. [8]). In this paper, a modification, in some respect similar to that in [8], will be given which is always practicable and the convergence of which is proved, too. The relation to the Remes-algorithm will be evident from a geometrical interpretation.

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Die vorliegende Arbeit ist eine gekürzte Fassung meiner Dissertation [5]. Herrn Prof. G. Hämmerlin danke ich herzlich für die Unterstützung bei deren Anfertigung.     

Die metrische Theorie der linearen Komplexbündel von Typ 1 des einfach isotropen RaumesJ 3 (1)

According toK. Strubecker ([18]–[21]) a three dimensional real affine space with the metricds ²=dx ²+dy ² is called a simply isotropic spaceJ ₃ ⁽¹⁾ . InJ ₃ ⁽¹⁾ exist 41 types of bundles of linear line complexes. In this paper we study the metric theory of a bundle of type 1. Especially we investigate the congruence of axis, we give some isotropic and affine results and we study the complementary bundle.

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Herrn em. o. Prof. Dr. W. Wunderlich zum 75. Geburtstag gewidmet

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Im folgenden beziehen wir uns stets auf die Bezeichnungen und Resultate aus [15].     

Ein stabiles Schichtenverfahren für allgemeine lineare Evolutionsgleichungen

Summary By generalizing Rothe's method of lines to initial value problemsu=A u, u(0)=u ₀ in Banach-spaces one obtains an approximation method that can be investigated within the theory of Lax and richtmyer [2]. It is well known that the generalized solutions of properly posed problems are semi-groups of class (C₀). In this paper the relation between the infinitesimal generator of that semi-group and the operatorA of the initial-value problem is discussed. As a consequence of the representation theorem of Hille-Yosida-Phillips one obtains that the method in question is stable iff the initial value problem is properly posed.

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Die Arbeit ist die gekürzte Fassung eines Teils meiner Dissertation [7], die ich am Lehrstuhl von Prof. G. Hämmerlin anfertigte. Ihm danke ich herzlich für die Unterstützung bei der Anfertigung der Arbeit.     

Propagation of discontinuity waves of any order through an elastic solid saturated with an inviscid fluid

Summary In this paper we study the propagation of discontinuity waves of any order through an elastic solid saturated with an inviscid fluid by using the linear theory derived in [1] by G. Szefer. By means of the singular surfaces theory we obtain the possible normal speeds of propagation of the wave front and the evolution law along the normal trajectories for transverse and longitudinal propagation.

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Riassunto In questo lavoro studiamo la propagazione di onde di discontinuità di ogni ordine in un mezzo elastico poroso saturato da un fluido non viscoso nell'ambito della teoria lineare formulata da G. Szefer in [1]. Usando il metodo delle superfici singolari otteniamo le possibili velocità normali di avanzamento del fronte d'onda e la legge di evoluzione delle discontinuità lungo le corrispondenti traiettorie normali sia per propagazione trasversale, sia per propagazione longitudinale.

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This work was supported by the G. N. F. M. of C. N. R.     

Über die Starrheit der n-Jacobi-Struktur

This article presents an elementary proof for the rigidity of the n-Jacobi structure, demonstrated by Serre [3]. Our proof is based upon ideas from Grothendieck [1].

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Meinem verehrten Lehrer Herrn Prof. Dr. K. Stein zum 60. Geburtstag gewidmet     

Approximation properties of modified Gamma operators

In this paper we present a survey of rates of pointwise approximation of modified Gamma operators Gn for locally bounded functions and absolutely continuous functions by using some inequalities and results of probability theory with the method of Bojanic-Cheng. In the paper a kind of locally bounded functions is introduced with different growth conditions in the fields of both ends of interval (0,+∞), and it is found out that the operators have different properties compared to the Gamma operators discussed in [X.M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl. 311 (2005) 389-401]. And we obtain two main theorems. Theorem 1 gives an estimate for locally bounded functions which subsumes the approximation of functions of bounded variation as a special case. Theorem 2 gives an estimate for absolutely continuous functions which is best possible in the asymptotical sense.     

Definitheitsbedingungen für relative Extrema bei Optimierungs- und Approximationsaufgaben

Summary In this paper, sufficient and necessary maximum conditions are established for a class of mathematical programming problems with an infinite set of restrictions, which is described by a finite number of inequalities. The criteria may be applied to nonlinear approximation problems and to the numerical solution of boundary value problems.

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Herrn Prof. Dr. Dr. h. c. L. Collatz zum 60. Geburtstag gewidmet     

Zur Approximation pseudoanalytischer Funktionen durch Pseudopolynome

In the theory of pseudoanalytic functions one can define (pseudoanalytic) rational functions, especially polynomials called “pseudopolynomials”. (See Bers [3], [4], Vekua [12]) Therefore it can be developed a theory of approximation and interpolation by rational functions. First results have been published by Bers [3] (Runge's theorem), Ismailov and Taglieva [8]. Let G be a domain of the complex plane bounded by a closed Jordan curve, let w(z) be pseudoanalytic in G. In this paper we deal with a relation between the behaviour of w(z) on C (Hölder-continuity) and the degree of approximation of w(z) by pseudopolynomials. The results correspond to certain theorems of Curtiss, Sewell and Walsh in the theory of analytic functions.     

Anelli Henseliani topologici - II

Summary In the present work we study some problems about henselian triples and henselization which have been introduced in our article [18]. Mainly we prove that henselian triples coincide with strong henselian triples and give a new formulation of the Hensel lemma, stronger than that we gave in [18]. Then we investigate some properties of henselian triples (changement of ideal or of topology, ecc.) and prove commutativity with quotient.

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Lavoro eseguito nell'ambito dei contratti di ricerca del Comitato Nazionale per la Matematica del C.N.R.

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Entrata in Redazione il 27 maggio 1971.     

Une classe de problèmes variationnels non linéaires de type elliptique ou parabolique

Summary In this paper we study existence and regularity properties for a class of degenerate elliptic and parabolic nonlinear variational problems.

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Entrata in Redazione il 22 aprile 1971.

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Boursier de la Fondation Calouste Gulbenkian (Portugal).     

Über die Extremaleigenschaft nichtlinearer interpolierender Splines

Summary In this paper we consider interpolation problems for nonlinear classes of spline functions. It is the aim to establish an extremal property for the solution of such a problem. Using this property, sufficient conditions for the existence of a solution of the interpolation problem are given. Some numerical examples illustrate the results.

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Die vorliegende Arbeit ist eine gekürzte Fassung des ersten Teils der Dissertation des Verfassers (Fakultät für Mathematik der Ludwig-Maximilians-Universität).     

Eine Bemerkung zum Satz von Müntz

Luxemburg and Korevaar obtain in [3] a relationship between entire functions and Müntz's approximation theorem. In this note we want to show that a slightly modified version of [2] implies the main results of [3].