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Journal of Function Spaces
Volume 2017 (2017), Article ID 9257076, 4 pages
https://doi.org/10.1155/2017/9257076
Letter to the Editor

Comment on “On the Frame Properties of Degenerate System of Sines”

Institute of Mathematics and Mechanics NAS of Azerbaijan, B. Vahabzade 9, 1141 Baku, Azerbaijan

Correspondence should be addressed to Aydin Sh. Shukurov

Received 26 May 2016; Accepted 8 December 2016; Published 15 March 2017

Academic Editor: Yoshihiro Sawano

Copyright © 2017 Aydin Sh. Shukurov. 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.

Abstract

The proof of Theorem 3.1 of the paper “On the Frame Properties of Degenerate System of Sines” (see (Bilalov and Guliyeva, 2012)) published earlier in this journal contains a gap; the reasoning given there to prove this theorem is not enough to state the validity of the mentioned theorem. To overcome this shortage we state the most general fact on the completeness of sine system which implies in particular the validity of this fact. It is shown in this note that the system , where is an exponential or trigonometric (cosine or sine) systems, becomes complete in the corresponding Lebesgue space or , respectively, whenever belongs to the corresponding Lebesgue space for all indices (under the evident natural condition ). It is also shown that the same conclusion does not remain valid for, in general, any complete or complete orthonormal system . Besides it, the largest class of functions for which the system is complete in space is determined.