Comment on “On the Frame Properties of Degenerate System of Sines”
Author
Source
Issue
Vol. 2017, Issue 2017 (31 Dec. 2017), pp.1-4, 4 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2017-03-15
Country of Publication
Egypt
No. of Pages
4
Main Subjects
Abstract EN
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)φn(t)}, where {φn(t)} is an exponential or trigonometric (cosine or sine) systems, becomes complete in the corresponding Lebesgue space Lp(-π,π) or Lp(0,π), respectively, whenever {ω(t)φn(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 {φn(t)}.
Besides it, the largest class of functions ω(t) for which the system {ωtsinnt}n∈N is complete in Lp(0,π) space is determined.
American Psychological Association (APA)
Shukurov, Aydin Sh.. 2017. Comment on “On the Frame Properties of Degenerate System of Sines”. Journal of Function Spaces،Vol. 2017, no. 2017, pp.1-4.
https://search.emarefa.net/detail/BIM-1176656
Modern Language Association (MLA)
Shukurov, Aydin Sh.. Comment on “On the Frame Properties of Degenerate System of Sines”. Journal of Function Spaces No. 2017 (2017), pp.1-4.
https://search.emarefa.net/detail/BIM-1176656
American Medical Association (AMA)
Shukurov, Aydin Sh.. Comment on “On the Frame Properties of Degenerate System of Sines”. Journal of Function Spaces. 2017. Vol. 2017, no. 2017, pp.1-4.
https://search.emarefa.net/detail/BIM-1176656
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1176656