Gyronormed Function Spaces

Document Type : Original Scientific Paper

Author

‎Department of Engineering, ‎Università degli Studi del Sannio, ‎Campania‎, ‎Italia

Abstract

‎In this paper‎, ‎we introduce a gyrodistance on the gyrolinear space of functions (whose gyronorm is measurable) from a measure space to the Möbius disk $\mathbb{D}$‎. ‎The gyrodistance in question can be expressed in terms of a modification of the Lebesgue integral‎, ‎which we will call the Lebesgue gyrointegral‎. ‎The gyronormed space generated in this manner‎, ‎which we will call the $L^1$ gyrospace‎, ‎is similar to the familiar $L^1$ function space in many aspects‎. ‎We establish several properties of the latter‎, ‎showing that many of them mirror those of classical $L^1$ spaces‎. ‎Finally‎, ‎we show that the gyrodistance in question induces a metric topology on the $L^1$ gyrospace‎.

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References

[1] A. A. Ungar, Thomas rotation and the parametrization of the Lorentz transformation group, Found. Phys. Lett. 1 (1988) 57- 89, https://doi.org/10.1007/BF00661317.
[2] A. A. Ungar, The relativistic noncommutative nonassociative group of velocities and the thomas rotation, Results Math. 16 (1989) 168-179, https://doi.org/10.1007/BF03322653.
[3] A. A. Ungar, The holomorphic automorphism group of the complex disk, Aequationes Math. 47 (1994) 240-254, https://doi.org/10.1007/BF01832962.
[4] A. A. Ungar, A Gyrovector Space Approach to Hyperbolic Geometry, Springer, 2009.
[5] A. A. Ungar, Thomas precession and its associated grouplike structure, Amer. J. Phys. 59 (1991) 824 -834, https://doi.org/10.1119/1.16730.
[6] A. A. Ungar, Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession, Springer, 2001.
[7] F. Santosa and W. W. Symes, Linear inversion of band-limited reflection seismograms, SIAM J. Sci. Statist. Comput. 7 (1986) 1307 - 1330, https://doi.org/10.1137/0907087.
[8] R. Tibshirani, Regression shrinkage and selection via the lasso, J. R. Statist. Soc. B 58 (1996) 267 - 288, https://doi.org/10.1111/j.2517-6161.1996.tb02080.x.
[9] S. Tian, Y. Yu and H. Guo, Variable selection and corporate bankruptcy forecasts, J. Bank. Finance 52 (2015) 89 - 100,
https://doi.org/10.1016/j.jbankfin.2014.12.003.
[10] M. Kern, R. Nagel and G. Palm, Dilations of positive operators: Construction and ergodic theory, Math. Z. 156 (1977) 265-277, https://doi.org/10.1007/BF01214414.
[11] M. A. Akcoglu, Positive contractions of L1 spaces, Math. Z. 143 (1975) 5-13, https://doi.org/10.1007/BF01173047.
[12] T. Abe and O. Hatori, Generalized gyrovector spaces and a Mazur-Ulam theorem, Publ. Math. Debrecen 87 (2015) 393 - 413, https://doi.org/10.5486/PMD.2015.7234.
[13] T. Abe, Normed gyrolinear spaces: a generalization of normed spaces based on gyrocommutative gyrogroups, Math. Interdisc. Res. 1 (2016) 143 - 172, https://doi.org/10.22052/mir.2016.13912.
[14] E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration and Hilbert Spaces, Princeton University Press, 2005.
Mathematics Interdisciplinary Research
Volume 10, Issue 3
September 2025
Pages 337-363

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