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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 717501, 13 pages
Research Article

On Singular Integrals with Cauchy Kernel on Weight Subspaces: The Basicity Property of Sines and Cosines Systems in Weight Spaces

1Nakhchivan State University, University Campus, AZ 7000 Nakhchivan, Azerbaijan
2Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, AZ 1141 Baku, Azerbaijan

Received 3 February 2011; Accepted 28 March 2011

Academic Editor: Shyam Kalla

Copyright © 2011 Tofig Isa Najafov and Saeed Farahani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A singular operator with Cauchy kernel on the subspaces of weight Lebesgue space is considered. A sufficient condition for a bounded action of this operator from a subspace to another subspace of weight Lebesgue space of functions is found. These conditions are not identical with Muckenhoupt conditions. Moreover, the completeness, minimality, and basicity of sines and cosines systems are considered.