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Free interpolation in H∞ and in some other function classes. I.

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Abstract

The first part of the survey is, basically, devoted to the description of the set of all the restrictions of the analytic functions of some class X onto a fixed subset E of their domain of definition; the problem consists in finding those sets E for which the corresponding class of restrictions is sufficiently simple. A tyical example is furnished by Carleson's interpolation theorem (X=H∞,E is a discrete subset of the open unit circle satisfying the well-known Newman-Carleson condition; in this case the class of restrictions coincides with the set of all functions bounded on E), to which the largest part of the paper is devoted. We also present the results of L. Carleson and P. Beurling on the linear operator of the interpolation of bounded functions by bounded analytic functions and on the estimate of the uniform norm of the interpolating functions.

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Authors
  1. S. A. Vinogradov
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  2. V. P. Khavin
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 47, pp. 15–54, 1974

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Vinogradov, S.A., Khavin, V.P. Free interpolation in H∞ and in some other function classes. I.. J Math Sci 9, 137–171 (1978). https://doi.org/10.1007/BF01578540

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