Abstract
In this study, we extend Beckner’s seminal work on the Fourier transform to the domain of Cayley–Dickson algebras, establishing a precise form of the Hausdorff–Young inequality for functions that take values in these algebras. Our extension faces significant hurdles due to the unique characteristics of the Cayley–Dickson Fourier transform. This transformation diverges from the classical Fourier transform in several key aspects: it does not conform to the Plancherel theorem, alters the interplay between derivatives and multiplication, and the product of algebra elements does not necessarily maintain the magnitude relationships found in classical settings. To overcome these challenges, our approach involves constructing the Cayley–Dickson Fourier transform by sequentially applying classical Fourier transforms. A pivotal part of our strategy is the utilization of a theorem that facilitates the norm-preserving extension of linear operators between spaces \(L^p\) and \(L^q.\) Furthermore, our investigation brings new insights into the complexities surrounding the Beckner–Hirschman Entropic inequality in the context of non-associative algebras.
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References
Albuquerque, H., Majid, S.: Quasialgebra structure of the octonions. J. Algebra 220(1), 188–224 (1999)
Babenko, K.I.: An inequality in the theory of Fourier integrals. Izv. Akad. Nauk SSSR Ser. Mat. 25, 531–542 (1961). [English transl., Amer. Math. Soc. Transl. (2) 44, 115–128]
Baez, J.C.: The octonions. Bull. Am. Math. Soc. (N.S.) 39(2), 145–205 (2002)
Beckner, W.: Inequalities in Fourier analysis. Ann. Math. (2) 102(1), 159–182 (1975)
Błaszczyk, Ł.: Octonion spectrum of 3D octonion-valued signals—properties and possible applications. In: 26th European Signal Processing Conference (EUSIPCO), Rome, Italy, pp. 509–513 (2018). https://doi.org/10.23919/EUSIPCO.2018.8553228
Błaszczyk, Ł: A generalization of the octonion Fourier transform to 3-D octonion-valued signals: properties and possible applications to 3-D LTI partial differential. Multidim. Syst. Sign. Process. 31, 1227–1257 (2020)
Cabrera, G.M., Rodríguez, P.A.: Non-associative Normed Algebras, vol. 1. The Vidav–Palmer and Gelfand–Naimark Theorems. Encyclopedia of Mathematics and Its Applications, vol. 154. Cambridge University Press, Cambridge (2014)
Cabrera, G.M., Rodríguez, P.A.: Non-associative Normed Algebras, vol. 2. Representation Theory and the Zel’manov Approach. Encyclopedia of Mathematics and Its Applications, vol. 167. Cambridge University Press, Cambridge (2018)
Ell, T.A., Bihan, N.L., Sangwine, S.J.: Quaternion Fourier Transforms for Signal and Image Processing, Hoboken. Focus Series in Digital Signal and Image Processing, Wiley/ISTE, Hoboken/London (2014)
Fan, S., Ren, G.: Fourier transform on Cayley-Dickson algebras (submitted)
Hirschman, I.I., Jr.: A note on entropy. Am. J. Math. 79, 152–156 (1957)
Huo, Q., Ren, G.: Structure of octonionic Hilbert spaces with applications in the Parseval equality and Cayley–Dickson algebras. J. Math. Phys. 63(4), Paper No. 042101 (2022)
Li, Y., Ren, G.: Real Paley–Wiener theorem for octonion Fourier transforms. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7513
Lian, P.: The octonionic Fourier transform: uncertainty relations and convolution. Signal Process. 164(2019), 295–300 (2019)
Lian, P.: Sharp Hausdorff–Young inequalities for the quaternion Fourier transforms. Proc. Am. Math. Soc. 148, 697–703 (2020)
Lieb, E.H.: Gaussian kernels have only Gaussian maximizers. Invent. Math. 102(1), 179–208 (1990)
Grafakos L.: Classical fourier analysis. In: Graduate Texts in Mathematics, vol. 249. Springer, New york (2014)
Mirzaiyan, Z., Esposito, G.: Generating rotating black hole solutions by using the Cayley–Dickson construction. Ann. Phys. 450, Paper No. 169223 (2023)
Mizoguchi, T., Yamada, I.: An algebraic translation of Cayley–Dickson linear systems and its applications to online learning. IEEE Trans. Signal Process. 62(6), 1438–1453 (2014)
Ren, G., Zhao, X.: The twisted group algebra structure of the Cayley–Dickson algebra. Adv. Appl. Clifford Algebras 33(4), Paper No. 49 (2023)
Restuccia, A., Sotomayor, A., Veiro, J.P.: A new integrable equation valued on a Cayley–Dickson algebra. J. Phys. A 51(34), 345203 (2018)
Snopek, K.M.: The study of properties of \(n\)-D analytic signals and their spectra in complex and hypercomplex domains. Radioengineering 21, 29–36 (2012)
Snopek, K.M.: New hypercomplex analytic signals and Fourier transforms in Cayley–Dickson algebras. Electron. Telecommun. Q. 55(3), 403–415 (2009)
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Fan, S., Ren, G. Hausdorff–Young Inequalities for Fourier Transforms over Cayley–Dickson Algebras. Adv. Appl. Clifford Algebras 34, 21 (2024). https://doi.org/10.1007/s00006-024-01326-x
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DOI: https://doi.org/10.1007/s00006-024-01326-x
Keywords
- Cayley–Dickson algebras
- Fourier transform
- Hausdorff–Young inequality
- Beckner–Hirschman entropic inequality