Abstract
In this article a function is constructed belonging to the class H 11 (S2) and having a singularity at a definite point on the sphere, as a consequence of which localization fails for the Laplace series of this function at the diametrically opposite point. The constructed example shows that the sufficient condition of localization in H p a of the spectral expansions in the class of all elliptic differential operators on an n-dimensional paracompact manifold cannot be improved (see [1]).
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Sh. A. Alimov, “On localization of spectral expansions,” Diff. Uravn.,10, No. 4, 744–746 (1974).
M. Gronwall, “On the degree of convergence of Laplace's series,” Trans. Am. Math. Soc.,15, No. 1, 1–30 (1914).
E. Kogbetliantz, “Analogues entre series spherique et series trigonometrique,” Ann. Sci. École Norm. Ser. 3,40, 292 (1923).
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Translated from Matematicheskie Zametki, Vol. 22, No. 4, pp. 517–523, October, 1977.
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Pulatov, A.K. Absence of localization of the Laplace series on the sphere for functions of the Nikol'skii class H 11 (S2). Mathematical Notes of the Academy of Sciences of the USSR 22, 779–783 (1977). https://doi.org/10.1007/BF01146423
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DOI: https://doi.org/10.1007/BF01146423