On Chebyshev subspaces in the space of multivariate differentiable functions

Kroó, András

doi:10.1090/S0002-9947-1985-0768745-8

On Chebyshev subspaces in the space of multivariate differentiable functions
HTML articles powered by AMS MathViewer

by András Kroó PDF

Trans. Amer. Math. Soc. 287 (1985), 839-852 Request permission

Abstract:

In the present paper we give a characterization of Chebyshev sub-spaces in the space of (real or complex) continuously-differentiable functions of two variables. We also discuss various applications of the characterization theorem.

References

B. Brosowski, Tschebyscheffsche Approximationen an differenzierbare Funktionen, Z. Angew. Math. Mech. 46 (1966), T39–T40 (German). MR 227681, DOI 10.1002/zamm.19660469015
Bruno Brosowski and Henry L. Loeb, Zur Eindeutigkeit der rationalen Tschebyscheff-Approximationen an stetig differenzierbare Funktionen, Numer. Math. 10 (1967), 51–55 (German). MR 216219, DOI 10.1007/BF02165159
Bruno Brosowski and Josef Stoer, Zur rationalen Tschebyscheff-Approximation differenzierbarer und analytischer Funktionen, Numer. Math. 12 (1968), 57–65 (German, with English summary). MR 240531, DOI 10.1007/BF02170997
L. Collatz, Approximation von Funktionen bei einer und bei mehreren unabhängigen Veränderlichen, Z. Angew. Math. Mech. 36 (1956), 198–211 (German, with English, French and Russian summaries). MR 83198, DOI 10.1002/zamm.19560360506
A. L. Garkavi, Dimensionality of polyhedra of best approximation for differentiable functions, Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959), 93–114 (Russian). MR 0104957
Alfred Haar, Die Minkowskische Geometrie und die Annäherung an stetige Funktionen, Math. Ann. 78 (1917), no. 1, 294–311 (German). MR 1511900, DOI 10.1007/BF01457106
A. N. Kolmogorov, A remark on the polynomials of P. L. Čebyšev deviating the least from a given function, Uspehi Matem. Nauk (N.S.) 3 (1948), no. 1(23), 216–221 (Russian). MR 0025609
András Kroó, On the unicity of best Chebyshev approximation of differentiable functions, Acta Sci. Math. (Szeged) 47 (1984), no. 3-4, 377–389. MR 783312
John C. Mairhuber, On Haar’s theorem concerning Chebychev approximation problems having unique solutions, Proc. Amer. Math. Soc. 7 (1956), 609–615. MR 79672, DOI 10.1090/S0002-9939-1956-0079672-3
T. J. Rivlin and H. S. Shapiro, Some uniqueness problems in approximation theory, Comm. Pure Appl. Math. 13 (1960), 35–47. MR 123127, DOI 10.1002/cpa.3160130104
Harold S. Shapiro, Topics in approximation theory, Lecture Notes in Mathematics, Vol. 187, Springer-Verlag, Berlin-New York, 1971. With appendices by Jan Boman and Torbjörn Hedberg. MR 0437981
Ivan Singer, Best approximation in normed linear spaces by elements of linear subspaces, Die Grundlehren der mathematischen Wissenschaften, Band 171, Publishing House of the Academy of the Socialist Republic of Romania, Bucharest; Springer-Verlag, New York-Berlin, 1970. Translated from the Romanian by Radu Georgescu. MR 0270044
V. I. Smirnov and N. A. Lebedev, Functions of a complex variable: Constructive theory, The M.I.T. Press, Cambridge, Mass., 1968. Translated from the Russian by Scripta Technica Ltd. MR 0229803
Hartmut Ehlich and Karl Zeller, Čebyšev-Polynome in mehreren Veränderlichen, Math. Z. 93 (1966), 142–143 (German). MR 198069, DOI 10.1007/BF01111031