Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Lukomskii S. F., Kruss I. S. Unitary extension principle on zero-dimensional locally compact groups. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 3, pp. 320-338. DOI: 10.18500/1816-9791-2023-23-3-320-338, EDN: AKZMKQ

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Unitary extension principle on zero-dimensional locally compact groups

Lukomskii Sergei Feodorovich, Saratov State University
Kruss Iuliia Sergeevna, Saratov State University

In this article, we obtain methods for constructing step  tight frames on an arbitrary locally compact zero-dimensional group. To do this, we use the principle of unitary extension. First, we indicate a method for constructing a step scaling function on an arbitrary zero-dimensional group. To construct the scaling function, we use an oriented tree and specify the conditions on the tree under which the tree generates the mask $m_0$ of a scaling function. Then we find conditions on the masks $m_0, m_1,\ldots  , m_q$ under which the corresponding wavelet functions $\psi_1, \psi_2,\ldots  ,\psi_q$ generate a tight frame. Using these conditions, we indicate a way of constructing such masks. In conclusion, we give examples of the construction of tight frames.

This work was supported by the Russian Science Foundation (project No. 22-21-00037, https://rscf.ru/project/22-21-00037/).
  1. Zhao H. (ed.). Mathematics in Image Processing. IAS/Park City Mathematics Series, 2013. Vol. 19. 245 p. https://doi.org/10.1090/pcms/019
  2. Ron A., Shen Z. Affine systems L2(Rd): The analysis of the analysis operator. Journal of Functional Analysis, 1997, vol. 148, iss. 2, pp. 408-447. https://doi.org/10.1006/jfan.1996.3079
  3. Farkov Y., Lebedeva E., Skopina M. Wavelet frames on Vilenkin groups and their approximation properties. International Journal of Wavelets, Multiresolution and Information Processing, 2015, vol. 13, iss. 5, 1550036 (19 p). https://doi.org/10.1142/S0219691315500368
  4. Shah F. A., Debnath L. Tight wavelet frames on local fields. Analysis, 2013, vol. 33, iss. 3, pp. 293–307. https://doi.org/10.1524/anly.2013.1217
  5. Ahmad O., Bhat M. Y., Sheikh N. A. Construction of Parseval framelets associated with GMRA on local fields of positive characteristic. Numerical Functional Analysis and Optimization, 2021, vol. 42, iss. 3, pp. 344–370. https://doi.org/10.1080/01630563.2021.1878370
  6. Albeverio S., Evdokimov S., Skopina M. p-adic multiresolution analysis and wavelet frames. Journal of Fourier Analysis and Applications, 2010, vol. 16, pp. 693–714. https://doi.org/10.1007/s00041-009-9118-5
  7. Lukomskii S. F. Multiresolution analysis on zero-dimensional Abelian groups and wavelets bases. Sbornik: Mathematics, 2010, vol. 201, iss. 5, pp. 669–691. https://doi.org/10.1070/SM2010v201n05ABEH004088
  8. Agaev G. N., Vilenkin N. Ya., Dzafarli G. M., Rubinstein A. I. Mul'tiplikativnye sistemy funkcij i garmonicheskij analiz na nul'mernykh gruppakh [Multiplicative Systems of Functions and Harmonic Analysis on Zero-Dimensional Groups]. Baku, Elm, 1981. 180 p. (in Russian).
  9. Albeverio S., Khrennikov A. Yu, Shelkovich V. M. Theory of p-adic Distributions: Linear and Nonlinear Models. Cambridge, Cambridge University Press, 2010. 351 p. https://doi.org/10.1017/CBO9781139107167
  10. Lukomskii S. F. Step refinable functions and orthogonal MRA on p-adic Vilenkin groups. Journal of Fourier Analysis and Applications, 2014, vol. 20, iss. 1, pp. 42–65. https://doi.org/10.1007/s00041-013-9301-6
  11. Lukomskii S., Vodolazov A. P-adic tight wavelet frames. 12 mar 2022. https://doi.org/10.48550/arXiv.2203.06352