Zur Eindeutigkeit der rationalen Tschebyscheff Approximationen an stetig differenzierbare Funktionen

Zusammenfassung In der vorliegenden Arbeit untersuchen wir die Aufgabe, eine stetig differenzierbare Funktion durch verallgemeinerte rationale Funktionen im Sinne vonTschebyscheff zu approximieren. Es wird ein Kriterium abgeleitet, das notwendig und hinreichend dafür ist, daß jede stetig differenzierbare Funktion höchstens eine Minimallösung besitzt.     

Zur Approximation analytischer Funktionen durch Polynome

Ohne Zusammenfassung     

Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

Zur Wertverteilung approximativ analytischer Funktionen

Ohne Zusammenfassung     

Zur verallgemeinerten rationalen Approximation

Ohne Zusammenfassung     

Zur Charakterisierung der positiven rationalen Funktionen

Ohne ZusammenfassungHerrn Professor Dr.Hubert Cremer zum 60. Geburtstag gewidmet     

Zur Abschätzung des Bestapproximationsfehlers bei der Approximation differenzierbarer Funktionen durch Polynome

The problem attacked here is to find smaller factorsγ (n,?),n > ?, for the inequality $$\delta _n [f] \leqq \gamma (n,\varrho ) \cdot \mathop {\sup }\limits_{x \in [ - 1,1]} |f^{(\varrho )} (x)|$$ than are already known. Heref(^? (x) denotes the?-th derivative off(x), andδ _n [f] is the error of the best Chebyshev-approximation off by algebraic polynomials of degree ≦n. A new approach to this problem is demonstrated and the results we got forn≦9,?≦8 by the use of a computer are presented.     

Gruppen und Lie-Algebren von rationalen Funktionen

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Abgeschlossene Garben differenzierbarer Funktionen

In this note we give a complete characterization of closed modulesheaves of N-differentiable functions of one variable in terms of distinguished systems of generating elements. For example closed pointwise finitely generated idealsheaves are those generated by one global analytic function in the case N=, by a constant in the case N<. Closed N-differentiable functions in the case N= are exactly those satisfying a Lojasiewicz-inequality. The several variable case however is remarkably different, as is pointed out.     

Über die Primideale differenzierbarer und integrierbarer Funktionen

Ohne Zusammenfassung     

Optimale hermite-interpolation differenzierbarer periodischer Funktionen

For the Favard class F_r in the space C_2π of continuous 2π-periodic functions we solve the following problem. Given x and knots x₀< x₁ < ··· < x_v−1., x_u− 2π we determine weights x_ki(0 k · n, 0 j < r) such that is minimal. The optimal weights are unique (except for a trivial case) and we obtain them from a system of periodic polynomial splines u_kj(0 k < n, 0 j< r): α_kj = u_kj(x). These splines induce an interpolation operator whose degree of approximation with respect to the class F_r is minimal if the knots are equidistant. Finally, we describe an efficient numerical procedure which shows how to compute the interpolation spline in the equidistant case.     

Darstellung reellvariabler und analytischer Funktionen durch verallgemeinerte Fourier- und Laplace-Integrale

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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.     

Ganze transzendente Funktionen mit rationalen Taylorkoeffizienten und vorgeschriebenen Nullstellen

Ohne Zusammenfassung     

Zur Existenz optimaler Quadraturformeln mit freien Knoten bei Integration analytischer Funktionen

Summary In the present paper we discuss the optimal quadrature rules for integration with positive continuous weight function in Hardy space H₂ of functions analytic in a circle of the complex plane. The new representations of the optimal weights and the norm of the error functional as functions of the nodes are obtained. On this basis we give an elementary proof for the existence of the optimal quadrature formula with free nodes.     

Defekte Werte eindeutiger analytischer Funktionen

Ohne Zusammenfassung Hellmuth Kneser zum 60. Geburtstage     

Über den Zusammenhang von Interpolation und diskreterT-Approximation mit rationalen Funktionen

Ohne Zusammenfassung