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More on free interpolation by functions which are regular outside a prescribed set

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Abstract

Let

, mE>0 It is shown that (even in the case when E is nowhere dense in

) there exist functions f, analytic in

and satisfying some strong additional conditions (for example, like: the Taylor series of the function f with center at the origin converges uniformly in the circle

, while the boundary values of the function

coincide with some function of the form

, where g ε C (T),

is the orthogonal projection from L2 onto h 2 ). Moreover, one establishes theorems on the free interpolation by such functions, showing that they are indeed “very many.”

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Literature cited

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    S. A. Vinogradov, “A refinement of Kolmogorov's theorem on the conjugate function and interpolational properties of uniformly convergent power series,” Tr. Mat. Inst. Akad. Nauk SSSR,155, 7–40 (1981).

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    S. V. Hruščev and S. A. Vinogradov, “Free interpolation in the space of uniformly convergent Taylor series,” in: Complex Analysis and Spectral Theory, Seminar, Leningrad (1979/80), Lecture Notes in Math., No. 864, 171–213 (1981).

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    S. V. Kislyakov, “The quantitative aspect of correction theorems,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,92, 182–191 (1979).

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    I. I. Privalov, Boundary Properties of Analytic Functions [in Russian], GITTL, Moscow-Leningrad (1950).

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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 107, pp. 71–88, 1982.

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Kislyakov, S.V. More on free interpolation by functions which are regular outside a prescribed set. J Math Sci 36, 342–352 (1987). https://doi.org/10.1007/BF01839606

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Keywords

  • Taylor Series
  • Additional Condition
  • Orthogonal Projection
  • Free Interpolation