English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Received: 20 July2017 Accepted: 17 Sep 2018 Extended Abstract Paper pages (237-252) Introduction ‎Frames for Hilbert spaces were first introduced by Duffin and‎ ‎Schaeffer in 1952 to study some problems in nonharmonic‎‎Fourier series‎, ‎reintroduced in 1986 by Daubechies‎, ‎Grossmann and‎ ‎Meyer‎. ‎Various generalizations of frames have been introduced and many applications of them in different branches have been presented‎. ‎Bessel multipliers in Hilbert spaces were introduced by Peter Balazs‎. ‎As we know in frame theory‎, ‎the composition of the synthesis and analysis operators of a frame is called the frame operator‎. ‎A multiplier for two Bessel sequences is an operator that combines the analysis operator‎, ‎a multiplication pattern with a fixed sequence‎, ‎called the symbol‎, ‎and the synthesis operator‎. ‎Bessel multipliers have useful applications‎, ‎for example they are used for solving approximation problems and they have applications as time-variant filters in acoustical signal processing‎. We mention that many generalizations of Bessel multipliers have been introduced, also multipliers have been studied for non-Bessel sequences. ‎Approximate duals in frame theory have important applications‎, ‎especially are used for the reconstruction of signals when it is difficult to find alternate duals‎. ‎Approximate duals are useful for wavelets‎, ‎Gabor systems and in sensor modeling‎. ‎Approximate duality of frames in Hilbert spaces was recently investigated by Christensen and Laugesen and some interesting applications of approximate duals were obtained‎. ‎For example‎, ‎it was shown that how approximate duals can be obtained via perturbation theory and some applications of approximate duals to Gabor frames especially Gabor frames generated by the Gaussian were presented‎. Afterwards, many authors studied approximate duals of Bessel sequences and many properties and generalizations of them were presented. In this note, we consider approximate duals for arbitrary sequences.

Approximate Duals Introduced by Multipliers and Their Role in the Reconstruction of Signals