Comment on "On the Frame Properties of Degenerate System of Sines". (15th March 2017) - Shukurov, Aydin Sh.
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 { ω ( t ) φ n ( t ) }, where { φ n ( t ) } is an exponential or trigonometric (cosine or sine) systems, becomes complete in the corresponding Lebesgue space L p ( - π, π ) or L p ( 0, π ), respectively, whenever { ω ( t ) φ n ( t ) } belongs to the corresponding Lebesgue space for all indices n (under the evident natural condition mes ⁡ { t : ω ( t ) = 0 } = 0 ). It is also shown that the same conclusion does not remain valid for, in general, any complete or complete orthonormal system { φ n ( t ) } . Besides it, the largest class of functions ω ( t ) for which the system { ω t sin ⁡ n t } n ∈ N is complete in L p ( 0, π ) space is determined.