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Version 2
This version is not peer-reviewed
Sums of A Pair of Orthogonal Frames
Ghanshyam Bhatt
Version 1
: Received: 9 May 2019 / Approved: 10 May 2019 / Online: 10 May 2019 (14:47:17 CEST)
Version 2 : Received: 30 June 2019 / Approved: 3 July 2019 / Online: 3 July 2019 (09:57:52 CEST)
Version 2 : Received: 30 June 2019 / Approved: 3 July 2019 / Online: 3 July 2019 (09:57:52 CEST)
A peer-reviewed article of this Preprint also exists.
Bhatt, G. Sums of A Pair of Orthogonal Frames. Mathematics 2019, 7, 582. Bhatt, G. Sums of A Pair of Orthogonal Frames. Mathematics 2019, 7, 582.
Journal reference: Mathematics 2019, 7, 582
DOI: 10.3390/math7070582
Abstract
Frames are more stable as compared to bases under the action of a bounded linear operator. Sums of different frames under the action of a bounded linear operator are studied with the help of analysis, synthesis and frame operators. A simple construction of frames from the existing ones under the action of such an operator is presented here. It is shown that a frame can be added to its alternate dual frames, yielding a new frame. It is also shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. Moreover, for a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate dual frames to be orthogonal. This allows for an easy construction of a large number of new frames.
Supplementary and Associated Material
Subject Areas
Frames; Orthogonal frames; Canonical dual; Alternate dual
Copyright: 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.
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