On Analysis and Synthesis Operators and Characterization of Synthesis Matrix of a Frame in Terms of Frame Operator

Authors

  • L. Njagi Department of Mathematics, Meru University of Science and Technology, P.O. Box 972-60200, Meru
  • B. M. Nzimbi School of Mathematics, University of Nairobi, Chiromo Campus, P. O. Box 30197-00100,Nairobi
  • S. K. Moindi School of Mathematics, University of Nairobi, Chiromo Campus, P. O. Box 30197-00100, Nairobi

DOI:

https://doi.org/10.53555/nnms.v5i12.524

Keywords:

Frame, analysis, synthesis, adjoint, Gramian, dual, operators

Abstract

In this research paper we introduce the operators associated with a frame. That is the Analysis and the Synthesis Operators and their basic properties. The structure of matrix representation of the Synthesis operator is also analysed. This matrix is what most frame constructions in fact focus on. The frame operator which is just the joining together of the analysis and synthesis operators is fundamental for the reconstruction of signals form frame coefficients. We also give a complete characterization of the synthesis matrix in terms of the frame operator.

References

Bodmann, B.G.: Optimal linear transmission by loss-insensitive packet encoding. Appl. Comput. Harmon. (2007)

Bodmann, B., Casazza, P.G.: The road to equal-norm Parseval frames. J. Funct. (2010)

Bodmann, B.G., Casazza, P.G., Kutyniok, G.: A quantitative notion of redundancy for finite frames. Appl. Comput. Harmon. (2011)

Bodmann, B.G., Casazza, P.G., Paulsen, V.I., Speegle, D.: Spanning and independence properties of frame partitions. Proc. Am. Math. Soc. (2012)

Bodmann, B., Lipshitz, S.: Randomly dithered quantization and sigma-delta noise shaping for finite frames. Appl. Comput. Harmon. (2008)

Bodmann, B.G., Paulsen, V.I.: Frames, graphs and erasures. Linear Algebra Appl. (2005).

Published

2018-12-31

Issue

Vol. 5 No. 12 (2018): Journal of Advance Research in Mathematics And Statistics (ISSN: 2208-2409)

Section

Research Article

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How to Cite

Njagi, L., Nzimbi, B. M., & Moindi, S. K. (2018). On Analysis and Synthesis Operators and Characterization of Synthesis Matrix of a Frame in Terms of Frame Operator. Journal of Advance Research in Mathematics and Statistics (ISSN 2208-2409), 5(12), 01-08. https://doi.org/10.53555/nnms.v5i12.524