Abstract
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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Literatur
Brosowski, B.: Nicht-lineare Tschebyscheff-Approximation. B. I. Mannheim 1968.
Burov, V. N.: The approximation of functions by polynomials satisfying nonlinear relations. Dokl. Akad. Nauk. SSSR138, 515–517 (1961).
Cheney, E. W.: Introduction to approximation theory. New York-St. Louis: McGraw-Hill 1966.
Deutsch, F.: On uniform approximation with interpolatory constraints. SIAM J. Appl. Math.24, 62–79 (1968).
Duffin, R. R., Karlovitz, L. A.: Formulation of linear programs in analysis I: Approximation theory. SIAM J. Appl. Math.16, 662–675 (1968).
Edwards, R. E.: Functional analysis, theory and applications. New York: Holt, Rinehart and Winston 1965
Gilormini, C.: Sur l'approximation par fractions rationnelles généralisées avec contraintes sur les coefficients. C. R. Acad. Sc. Paris265, 253–256 (1967a).
——: Approximation par fractions rationnelles généralisées dont les coefficients vérifient des relations lineaires. C. R. Acad. Sc. Paris264, 795–798 (1967b).
Hoffmann, K.-H.: Über nichtlineare Tschebyscheff-Approximation mit Nebenbedingungen. Inaugural-Diss., München (1968).
Jones, R. C., Karlovitz, L. A.: Iterative construction of constrained Chebyschev approximation of continuous functions. SIAM J. Num. Anal. B5, 574–585 (1968).
Kelly, J. L., Namioka, I.: Linear topological spaces. Princeton, New Jersey: D. van Nostrand Company, Inc. 1963
Köthe, G.: Topologische lineare Räume I. Berlin-Göttingen-Heidelberg: Springer 1960.
Laurent, P. J.: Charakterisierung und Gewinnung einer besten Approximation in einer konvexen Teilmenge eines normierten Raumes. Vortrag, gehalten auf der Tagung „Funktionalanalytische Methoden der numerischen Mathematik”, Oberwolfach 1967.
—— Theoremes de caracterisation on approximation convexe. Mathematica (33),1, 95–111 (1968).
Markov, V. A.: Über Polynome, die in einem gegebenen Intervall möglichst wenig von Null abweichen. Math. Annalen77, 213–258 (1916).
Paszkowski, S.: Sur l'approximation uniforme avec des noeuds. Ann. Polon. Math.2, 118–135 (1955)
Rice, J. R.: Approximation with convex constraints. SIAM J. Appl. Math.11, 15–32 (1963)
Schumaker, L. L., Taylor, G. D.: On approximation by polynomials having restricted ranges. MRC Tech. Summ. Report 835 (1968).
Taylor, G. D.: On approximation by polynomials having restricted ranges. SIAM J. Num. Anal. B5, 258–268 (1968).
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Erster Teil einer gekürzten Fassung der Dissertation des Verfassers [1968].
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Hoffmann, KH. Zur Theorie der nichtlinearen Tschebyscheff-Approximation mit Nebenbedingungen. Numer. Math. 14, 24–41 (1969). https://doi.org/10.1007/BF02165097
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DOI: https://doi.org/10.1007/BF02165097