Abstract
We prove that, on a convex polygon, there exist functions from the Smirnov classE whose series of exponents diverge in the metric of the spaceE. Similar facts are established for the convergence almost everywhere on the boundary of a polygon, for the uniform convergence on a closed polygon, and for the pointwise convergence at noncorner points of the boundary.
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References
I. I. Privalov,Boundary Properties of Analytic Functions [in Russian], Gostekhizdat, Moscow-Leningrad (1950).
A. F. Leont'ev,Series of Exponents [in Russian], Nauka, Moscow (1976).
A. M. Sedletskii, “Exponential bases in spacesE p on convex polygons,”Izv. Akad. Nauk SSSR, Ser. Mat.,42, No. 5, 1101–1119 (1978).
Yu. I. Mel'nik, “On representation of regular functions by Dirichlet series on closed convex polygons,”Ukr. Mat. Zh.,29, No. 6, 826–830 (1977).
Yu. I. Mel'nik, “On the rate of convergence of series of exponents representing functions regular on convex polygons,”Ukr. Mat. Zh.,37, No. 6, 719–722 (1985).
Yu. I. Mel'nik, “On the uniqueness of expansion of functions regular on convex polygons into series of exponents,”Ukr. Mat. Zh.,34, No. 2, 217–219 (1982).
A. Zygmund,Trigonometric Series [Russian translation], Vol. 1, Mir, Moscow (1965).
Additional information
Deceased.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 443–445, April, 1994.
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Mel'nik, Y.I. On divergence of series of exponents representing functions regular in convex polygons. Ukr Math J 46, 471–474 (1994). https://doi.org/10.1007/BF01060419
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DOI: https://doi.org/10.1007/BF01060419