Abstract
Octonions are 8-dimensional hypercomplex numbers which form the biggest normed division algebras over the real numbers. Motivated by applications in theoretical physics, continuous octonionic analysis has become an area of active research in recent year. Looking at possible practical applications, it is beneficial to work directly with discrete structures, rather than approximate continuous objects. Therefore, in previous papers, we have proposed some ideas towards the discrete octonionic analysis. It is well known, that there are several possibilities to discretise the continuous setting, and the Weyl calculus approach, which is typically used in the discrete Clifford analysis, to octonions has not been studied yet. Therefore, in this paper, we close this gap by presenting the discretisation of octonionic analysis based on the Weyl calculus.
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References
Baez, J.: The octonions. Bull. Am. Math. Soc. 39, 145–205 (2002)
Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis. In: Pitman Research Notes in Mathematics, vol. 76. Boston (1982)
Brackx, F., De Schepper, H., Sommen, F., Van de Voorde, L.: Discrete Clifford analysis: a germ of function theory. In: Hypercomplex Analysis, Birkhäuser Basel, pp. 37–53 (2009)
Burdik, C., Catto, S., Gürcan, Y., Khalfan, A., Kurt, L., La Kato, V.: \(SO(9,1)\) group and examples of analytic functions. J. Phys.: Conf. Ser. 1194, 012016 (2019)
Cerejeiras, P., Faustino, N., Vieira, N.: Numerical Clifford analysis for nonlinear Schrödinger problem. Numer. Methods Partial Differ. Eq. 24(4), 1181–1202 (2008)
Cerejeiras, P., Kähler, U., Ku, M., Sommen, F.: Discrete Hardy spaces. J. Fourier Anal. Appl. 20(4), 715–750 (2014)
Cerejeiras, P., Kähler, U., Legatiuk, A., Legatiuk, D.: Boundary values of discrete monogenic functions over bounded domains in \(\mathbbm {R}^{3}\). In: Alpay, D., Vajiac, M. (eds.) Linear Systems, Signal Processing and Hypercomplex Analysis. Operator Theory: Advances and Applications, vol. 275, pp. 149–165. Birkhäuser (2020)
Cerejeiras, P., Kähler, U., Legatiuk, A., Legatiuk, D.: Discrete Hardy spaces for bounded domains in \(\mathbbm {R}^{n}\). Complex Anal. Oper. Theory 15, 4 (2021)
Constales, D., Kraußhar, R.S.: Octonionic Kerzman-Stein operators. Complex Anal. Oper. Theory 15, 104 (2021)
Faustino, N., Kähler, U.: Fischer decomposition for difference Dirac operators. Adv. Appl. Clifford Algebras 17, 37–58 (2007)
Faustino, N., Kähler, U., Sommen, F.: Discrete Dirac operators in Clifford analysis. Adv. Appl. Clifford Algebras 17(3), 451–467 (2007)
Frenod, E., Ludkowski, S.V.: Integral operator approach over octonions to solution of nonlinear PDE. Far East J. Math. Sci. (2017)
Gogberashvili, M.: Octonionic geometry and conformal transformations. Int. J. Geomet. Methods Mod. Phys. 13(7), 1650092 (2016)
Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley & Sons, Chichester, New York (1997)
Gürlebeck, K., Hommel, A.: On finite difference Dirac operators and their fundamental solutions. Adv. Appl. Clifford Algebras 11, 89–106 (2001)
Gürlebeck, K., Habetha, K., Sprößig, W.: Application of Holomorphic Functions in Two and Higher Dimensions. Birkhäuser, Basel (2016)
Kauhanen, J., Orelma, H.: Cauchy-Riemann operators in octonionic analysis. Adv. Appl. Clifford Algebras 28, 1 (2018)
Kauhanen, J., Orelma, H.: On the structure of octonion regular functions. Adv. Appl. Clifford Algebras 29, 77 (2019)
Kraußhar, S., Legatiuk, A., Legatiuk, D.: Towards discrete octonionic analysis. In: Vasilyev, V. (eds.) Differential Equations, Mathematical Modeling and Computational Algorithms. DEMMCA 2021. Springer Proceedings in Mathematics and Statistics, vol. 423, pp. 51–63. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-28505-9_4
Kraußhar, S., Legatiuk, D.: Cauchy formulae and Hardy spaces in discrete octonionic analysis. In: Complex Analysis and Operator Theory (2023). (Accepted for Publication)
Lasenby, A.N.: Some recent GA results in mathematical physics and the GA approach to the fundamental forces of nature. Presentation at AGACSE 2021, YouTube Video (2021). https://www.youtube.com/watch?v=fFj4E7q4hbY
Najarbashi, G., Seifi, B., Mirzaei, S.: Two- and three-qubit geometry, quaternionic and octonionic conformal maps, and intertwining stereographic projection. Quant. Inf. Process. 15, 509–528 (2016)
Huo, Q., Ren, Q.: Structure of octonionic Hilbert spaces with applications in the Parseval equality and Cayley-Dickson algebras. J. Math. Phys. 63(4), 042101 (2022)
Li, X.-M., Peng, L.-Z.: Three-line theorems on the octonions. Acta Math. Sinica 20(3), 483–490 (2004)
Li, X.-M., Peng, L.-Z., Qian, T.: Cauchy integrals on Lipschitz surfaces in octonionic space. J. Math. Anal. Appl. 343, 763–777 (2008)
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Kraußhar, R.S., Legatiuk, D. (2024). Weyl Calculus Perspective on the Discrete Stokes’ Formula in Octonions. In: Sheng, B., Bi, L., Kim, J., Magnenat-Thalmann, N., Thalmann, D. (eds) Advances in Computer Graphics. CGI 2023. Lecture Notes in Computer Science, vol 14498. Springer, Cham. https://doi.org/10.1007/978-3-031-50078-7_29
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