Abstract
The boundedness of the Faber operator T and its inverse T−1, considered as mappings between various spaces of functions, is discussed. The relevance of this to problems of approximation, by polynomials or by rational functions, to functions defined on certain compact subsets of ¢ is explained.
Keywords
- Rational Approximation
- Besov Space
- Jordan Curve
- Supremum Norm
- Hankel Operator
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The author thanks the Department of Mathematics of the University of California at San Diego for its kind hospitality while this report was being written.
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References
Al'per, S. Ya., On the uniform approximation to functions of a complex variable on closed domains (in Russian), Izv. Akad. Nauk. S.S.S.R. Ser. Mat., 19 (1955), 423–444.
Anderson, J. M., and Clunie, J., Isomorphisms of the disc algebra and inverse Faber sets, to appear.
Andersson, J.-E., "On the degree of polynomial and rational approximation of holomorphic functions," Dissertation, Univ. of Göteborg, 1975.
Andersson, J.-E., On the degree of weighted polynomial approximation of holomorphic functions, Analysis Mathematica, 2 (1976), 163–171.
Andersson, J.-E., On the degree of polynomial approximation in EP(D), J. Approx. Theory, 19 (1977), 61–68.
Brudny, Ju. A., Spaces defined by means of local approximations, Trans. Moscow Math. Soc., 24 (1971).
Brudny, Ju. A., Rational approximation and imbedding theorems, Soviet Math. Dokl., 20 (1979), 681–684.
Duren, P. L., Theory of H P-spaces, Academic Press, New York, 1970.
Dynkin, E. M., A constructive characterization of the Sobolev and Besov classes, Proc. Steklov Inst. Math. (1983), 39–74.
Faber, G., Über Polynomische Entwicklungen, Math. Ann., 57 (1903), 389–408.
Koosis, P., Introduction to HP spaces, London Math. Soc. Lecture Note Series, 40, Cambridge University Press, 1980.
Kövari, T., On the order of polynomial approximation for closed Jordan domains, J. Approx. Theory, 5 (1972), 362–373.
Kövari, T., and Pommerenke, Ch., On Faber polynomials and Faber expansions, Math. Zeitschr., 99 (1967), 193–206.
Peller, V. V., Hankel operators of class 212D000p and their applications (rational approximation, Gaussian processes, the problem of majorizing operators), Math. U.S.S.R. Sbornik, 41 (1982), 443–479.
Peller, V. V., Rational approximation and the smoothness of functions, Zap. Nauch. Sem. L.O.M.I., 126 (1983), 150–159.
Rochberg, R., Trace ideal criteria for Hankel operators and commutators, Indiana Univ. Math. J., 31 (1982), 913–925.
Semmes, S., Trace ideal criteria for Hankel operators, 0 < p < 1, preprint.
Stein, E. M., Singular Integrals and Differentiability Properties of functions, Princeton Univ. Press, Princeton, N.J., 1970.
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© 1984 Springer-Verlag
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Anderson, J.M. (1984). The faber operator. In: Graves-Morris, P.R., Saff, E.B., Varga, R.S. (eds) Rational Approximation and Interpolation. Lecture Notes in Mathematics, vol 1105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072396
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DOI: https://doi.org/10.1007/BFb0072396
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