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Spaces of analytic functions in a region with an angle

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In this paper we consider the space Ap of analytic functions which are p-power integrable in a region with an angle. We find a set of numbers p and q (1/p+1/q=1) (which depend on the magnitude of the angle) for which the spaces Ap and Aq are mutually conjugate. In each of these spaces we introduce the orthonormal system

$$e_n = \sqrt {{{\left( {n + 1} \right)} \mathord{\left/ {\vphantom {{\left( {n + 1} \right)} \pi }} \right. \kern-\nulldelimiterspace} \pi }} \varphi \prime \varphi ^n ,n = 0,1, \ldots ,$$

whereϕ is the conformal mapping of the region onto the unit disc. We prove it is dense and determine when it will be a basis.

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Literature cited

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    V. P. Zakharyuta and V. I. Yudovich, “The general form of a linear functional on H′p,” Uspekhi Matem. Nauk,19, No. 2, 139–142 (1964).

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    K. Yosida, Functional Analysis [in Russian], Mir, Moscow (1967).

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    M. A. Evgrafov, Analytic Functions [in Russian], Nauka, Moscow (1968).

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    Mathematical Reference Library, Functional Analysis [in Russian], Nauka, Moscow (1964).

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Translated from Matematicheskie Zametki, Vol. 18, No. 3, pp. 411–420, September, 1975.

In conclusion I wish to thank S. G. Kreyn for a discussion of these results.

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Shikhvatov, A.M. Spaces of analytic functions in a region with an angle. Mathematical Notes of the Academy of Sciences of the USSR 18, 833–839 (1975).

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  • Analytic Function
  • Unit Disc
  • Conformal Mapping
  • Orthonormal System