Rocky Mountain Journal of Mathematics

Galois wavelet transforms over finite fields

Arash Ghaani Farashahi

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Abstract

In this article, we introduce the abstract notion of Galois wavelet groups over finite fields as the finite group of Galois dilations, and translations. We then present a unified theoretical linear algebra approach to the theory of Galois wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as a finite coherent sum of Galois wavelet coefficients as well.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 1 (2019), 79-99.

Dates
First available in Project Euclid: 10 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1552186953

Digital Object Identifier
doi:10.1216/RMJ-2019-49-1-79

Mathematical Reviews number (MathSciNet)
MR3921868

Zentralblatt MATH identifier
07036620

Subjects
Primary: 12E20: Finite fields (field-theoretic aspects) 42C40: Wavelets and other special systems
Secondary: 12F10: Separable extensions, Galois theory 13B05: Galois theory 20G40: Linear algebraic groups over finite fields 81R05: Finite-dimensional groups and algebras motivated by physics and their representations [See also 20C35, 22E70]

Keywords
Finite field Galois wavelet group Galois wavelet representation Galois wavelet transform Galois dilation operator periodic (finite size) data prime integer

Citation

Farashahi, Arash Ghaani. Galois wavelet transforms over finite fields. Rocky Mountain J. Math. 49 (2019), no. 1, 79--99. doi:10.1216/RMJ-2019-49-1-79. https://projecteuclid.org/euclid.rmjm/1552186953


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References

  • A. Arefijamaal and R.A. Kamyabi-Gol, On the square integrability of quasi regular representation on semidirect product groups, J. Geom. Anal. 19 (2009), 541–552.
  • A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comp. Harmon. Anal. 35 (2013), 535–540.
  • ––––, Image processing by alternate dual Gabor frames, Bull. Iranian Math. Soc. 42 (2016), 1305–1314.
  • L. Cohen, Time-frequency analysis, Prentice-Hall, New York, 1995.
  • H.G. Feichtinger, W. Kozek and F. Luef, Gabor analysis over finite Abelian groups, Appl. Comp. Harmon. Anal. 26 (2009), 230–248.
  • K. Flornes, A. Grossmann, M. Holschneider and B. Torrésani, Wavelets on discrete fields, Appl. Comp. Harmon. Anal. 1 (1994), 137–146.
  • G.B. Folland, A course in abstract harmonic analysis, CRC Press, Boca Raton, 1995.
  • H. Führ, Abstract harmonic analysis of continuous wavelet transforms, Springer-Verlag, Berlin, 2005.
  • A. Ghaani Farashahi, Cyclic wave packet transform on finite Abelian groups of prime order, Int. J. Wavelets Multiresolut. Inf. Proc. 12 (2014), 1450041.
  • ––––, Wave packet transform over finite fields, Electr. J. Lin. Alg. 30 (2015), 507–529.
  • ––––, Classical coherent state transforms over finite fields, Inter. J. Math. Game Th. Alg. 25 (2016), 273–297.
  • ––––, Wave packet transforms over finite cyclic groups, Lin. Alg. Appl. 489 (2016), 75–92.
  • ––––, Theoretical frame properties of wave-packet matrices over prime fields, Lin. Multilin. Alg. 65 (2017), 2508–2529.
  • ––––, Structure of finite wavelet frames over prime fields, Bull. Iranian Math. Soc. 43 (2017), 109–120.
  • G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, Oxford University Press, New York, 1979.
  • C.P. Johnston, On the pseudodilation representations of Flornes, Grossmann, Holschneider, and Torrésani, J. Fourier Anal. Appl. 3 (1997), 377–385.
  • J.B. Lima and R.M. Campello de Souza, Fractional cosine and sine transforms over finite fields, Lin. Alg. Appl. 438 (2013), 3217–3230.
  • R.J. McEliece, Finite fields for computer scientists and engineers, Springer Inter. Eng. Comp. Sci. (1987).
  • G.L. Mullen and D. Panario, Handbook of finite fields, Discr. Math. Appl., Chapman and Hall/CRC, 2013.
  • G. Pfander, Gabor frames in finite dimensions, Finite frames, theory and applications, in Finite frames, Applied and numerical harmonic analysis, Birkhauser, Boston, 2013.
  • O. Pretzel, Error-correcting codes and finite fields, Oxford Appl. Math. Comp. Sci. (1996).
  • D. Ramakrishnan and R.J. Valenza, Fourier analysis on number fields, Springer-Verlag, New York, 1999.
  • R. Reiter and J.D. Stegeman, Classical harmonic analysis, Oxford University Press, New York, 2000.
  • H. Riesel, Prime numbers and computer methods for factorization, Birkhauser, Boston, 1994.
  • A. Vourdas, Harmonic analysis on a Galois field and its subfields, J. Fourier Anal. Appl. 14 (2008), 102–123.