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.
Highlights
Frames are alternatives to a Riesz or orthonormal basis in Hilbert spaces
The orthogonality of a pair of frames plays an important role in this setting and we characterize the orthogonality of alternate dual frames in order to obtain new frames as a sum
We show that a frame can be added to any of its alternate dual frames to yield a new frame
Summary
Frames are alternatives to a Riesz or orthonormal basis in Hilbert spaces. Frame theory plays an important role in signal processing, image processing, data compression and many other applied areas. The analysis operator is the Hilbert space adjoint operator to the synthesis operator These operators are well defined and bounded because X is a Bessel sequence [1] (Lemma 5.2.1). ∑ h f , yj ixj j ∈J is an identity, the Bessel sequences X and Y are frames and are called dual frames [6]. In this case, the reconstruction formula takes the form f =. We provide an easy proof of this through the use of analysis and synthesis operators (Theorem 1) This improves the result presented in [8] (Proposition 3.1).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
