Deundyak V.M., Leonov D.A., Senchukova A.A. Symbolic Calculation and Invertibility of Convolution Operators on the Infinite Dihedral Group

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https://doi.org/10.15688/mpcm.jvolsu.2020.3.6

Vladimir M. Deundyak
Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Algebra and Discrete Mathematics,
Southern Federal University,
Institute of Mathematics, Mechanics and Computer Science named after I.I. Vorovich
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Milchakova St, 8a, 344090 Rostov-on-Don, Russian Federation;Senior Researcher,
FGNU NII "Specvuzavtomatika"
Gazetniy Lane, 51, 344002 Rostov-on-Don, Russian Federation

Dmitriy A. Leonov
Postgraduate Student, Department of Algebra and Discrete Mathematics,
Southern Federal University,
Institute of Mathematics, Mechanics and Computer Science named after I.I. Vorovich
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Milchakova St, 8a, 344090 Rostov-on-Don, Russian Federation

Angelina A. Senchukova
Postgraduate Student, Department of Algebra and Discrete Mathematics,
Southern Federal University,
Institute of Mathematics, Mechanics and Computer Science named after I.I. Vorovich
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Milchakova St, 8a, 344090 Rostov-on-Don, Russian Federation

Abstract. Nowadays, convolution operators on discrete noncommutative groups are under intensive research due to their applications, in particular, in the theory and practice of data networking, in image analysis, and in problems of diffraction by bodies with a noncommutative symmetry group. The symbolic calculation for algebra of convolution equations on the noncommutative infinite dihedral group D inf has been developed. Necessary and sufficient conditions of invertibility of convolution operators from this algebra in terms of symbolic calculation have been found in this paper. Besides, inclosure of algebra of convolution equations on D inf into matrix algebra of convolution operators on the group of whole numbers extended with involutive operator has been constructed.

In the theory of projection methods of the solution of operator equations the sequence of equations with more simple operators is constructed in order to approximate the solution of original equation with some accuracy, i.e. the reduction of original invertible operator to a more simple invertible operator. The connection between dual object of D inf and finite noncommutative dihedral group  D m is studied. On the basis of this the operator of reduction that maps invertible operator of convolution on D inf to invertible convolution operator on D m is constructed in this paper.

Key words: convolution operator, finite noncommutative dihedral group, inifinite noncommutative dihedral group, Fourier transformation, dual object, invertibility of convolution operator.

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Symbolic Calculation and Invertibility of Convolution Operators on the Infinite Dihedral Group by Deundyak V.M., Leonov D.A., Senchukova A.A. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. Vol. 23 No. 3 2020, pp. 60-75

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