Fourier Method for Solving Two-Sided Convolution Equations on Finite Noncommutative Groups

• Published in: Computational mathematics and mathematical physics Vol. 58; no. 10; pp. 1562 - 1572
• Main Authors: Deundyak, V. M., Leonov, D. A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-10-2018; Springer Nature B.V
• Subjects: Algorithms; Complexity; Computation; Computational Mathematics and Numerical Analysis; Convolution; Fourier transforms; Group theory; Mathematical analysis; Mathematics; Mathematics and Statistics; Operators (mathematics); Quantum mechanics; two-sided convolution operators; finite noncommutative groups; two-sided convolution equations; finite Heisenberg group; dihedral group; fast Fourier transform
• ISSN: 0965-5425; 1555-6662
• DOI: 10.1134/S0965542518100044

The Fourier method on commutative groups is used in many fields of mathematics, physics, and engineering. Nowadays, this method finds increasingly wide application to non-commutative groups. Along with the one-sided convolution operators and the corresponding convolution equations, two-sided convolution operators on noncommutative groups are studied. Two-sided convolution operators have a number of applications in complex analysis and are used in quantum mechanics. In this paper, two-sided convolutions on arbitrary finite noncommutative groups are considered. A criterion for the inversibility of the two-sided convolution operator is obtained. An algorithm for solving the two-sided convolution equation on an arbitrary finite noncommutative group, using the Fourier transform, is developed. Estimates of the computational complexity of the algorithm developed are given. It is shown that the complexity of solving the two-sided convolution equation depends both on the type of the group representation and on the computational complexity of the Fourier transform. The algorithm is considered in detail on the example of the finite dihedral group
and the Heisenberg group
over a simple Galois field, and the results of numerical experiments are presented.