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Discrete orthogonal transforms on lattices of integer elements of quadratic fields
V.M. Chernov 1,2

IPSI RAS – Branch of the FSRC "Crystallography and Photonics" RAS,
443001, Samara, Russia, Molodogvardeyskaya 151,
Samara National Research University, 443086, Samara, Russia, Moskovskoye Shosse 34

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DOI: 10.18287/2412-6179-CO-809

Pages: 142-148.

Full text of article: Russian language.

In this paper, we introduce a new class of discrete orthogonal transforms (DОT) defined on lattices of integer elements of quadratic fields. The method of synthesis of such transforms essentially uses the specifics of the representation of integer quadratic elements in the so-called quasi-canonical number systems. This article, which presents the results of the first part of the author's research, deals exclusively with problems related to binary number systems in quadratic fields. We also consider the issues of synthesis of fast algorithms of the introduced and the possibility of their application to the analysis of fractal (or self-similar) objects. We also consider the issues of synthesis of fast algorithms of the introduced methods and the possibility of their application for the analysis of fractal (or self-similar) objects.

discrete orthogonal transformations, number systems, quadratic fields, machine arithmetic.

Chernov VM. Discrete orthogonal transformations on lattices of integer elements of quadratic fields. Computer Optics 2021; 45(1): 142-148. DOI: 10.18287/2412-6179-CO-809.

The work was partly funded by the Russian Federation Ministry of Science and Higher Education within a state contract with the "Crystallography and Photonics" Research Center of the RAS under agreement 007-ГЗ/Ч3363/26 in part of «number systems» and by Russian Foundation for Basic Research (Grants 19-07-00357 А and 18-29-03135_ мк) in part of "machine arithmetic".


  1. Franks LE. Signal theory. Prentice-Hall Inc; 1969.
  2. Agayan S. Successes and problems of orthogonal transformations for signal-image processing. In Book: Zhuravlev YuI, ed. Pattern recognition. Classification. Forecasting. Mathematical tecniques and their application [In Russian]. Moscow: "Nauka" Publisher; 1990: 246-215.
  3. Chernov VM. Discrete orthogonal transforms with bases generated by self-similar sequences. Computer Optics 2018; 42(5): 904-911. DOI: 10.18287/2412-6179-2018-42-5-904-911.
  4. Chernov VM. On a class of Dirichlet series with finite Lindelef functions. Research on Number Theory 1982; 8: 92-95
  5. Chernov VM. Some spectral properties of fractal curves. Mach Graph Vis 1996; 5(1/2): 413-422.
  6. Chernov VM. Tauber theorems for Dirichlet series and fractals. Proc 13th ICPR 1996; 2: 656-661. DOI: 10.1109/ICPR.1996.546905.
  7. Schroeder M. Fractal, chaos, power laws: Minutes from an infinite paradise. New York: WH Freeman&Co; 1991: 456.
  8. Varichenko LV, Labunets VG, Rakov MA. Abstract algebraic systems and digital signal processing [In Russian]. Kyiv: “Naukova Dumka” Publisher; 1986.
  9. Soifer VA, Kupriyanov AV. Analysis and recognition of the nanoscale images: conventional approach and novel problem statement. Computer Optics 2011; 35(2):136-144.
  10. Kasparyan M, Chernov V. Discrete cosine transform on pre-fractal domains. Proc 2nd International Eurasian Conference on Mathematical Sciences and Applications (IECMSA-2013) 2013: 431.
  11. Chernov VM, Kasparyan MS. Discrete orthogonal transforms on fundamental domains of canonical number systems. Computer Optics 2013; 37(4): 484-488.
  12. Katai I, KovacsB. Canonical number systems in imaginary quadratic fields. Acta Mathematica Hungarica 1981; 37: 159-164.
  13. Bogdanov PS, Chernov VM. Classification of binary quasicanonical number systems in imaginary quadratic fields. Computer Optics 2013; 37(3): 391-400.
  14. Thuswardner J. Elementary properties of canonical numder systems inquadratic fields. In Book: Bergum GE, Philippou A, Horadam AF, eds. Application of Fibonacci numbers. Vol 7. Dordrecht: Springer Science+Business Media; 1998: 405-414.
  15. Wang R. Introduction to orthogonal transforms: With applications in data processing and analysis. Cambridge: Cambridge University Press; 2012.
  16. Borevich ZI, Shafarevich IR. Number theory. New York, London: Academic Press; 1966.
  17. Nussbaumer HJ. Fast Fourier transform and convolution algorithms. Berlin, Heidelberg: Springer-Verlag; 1982.
  18. Blahut RE. Fast algorithms for digital signal processing. Reading, MA: Addison-Wesley Publishing Company Inc; 1985.
  19. Chernov VM. Arithmetic methods of fast algorithm of discrete orthogonal transforms synthesis. Moscow: "Fizmathlit" Publisher; 2007.
  20. Chernov VM, Chicheva MA. Discrete orthogonal transforms on multisets associated with complete sequences. Trudy Instituta Matematiki i Mekhaniki URO RAN 2020; 26(3): 249-257.

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