Hidden subspace algorithm in white noise analysis
HTML articles powered by AMS MathViewer
- by Jeremy J. Becnel PDF
- Trans. Amer. Math. Soc. 364 (2012), 5035-5055 Request permission
Abstract:
In 2003, Professors Lomonaco and Kauffman developed an algorithm to find a hidden subspace of a functional. The algorithm was developed in the spirit of Feynman path integrals but lacked mathematical rigor. In this paper we make use of the framework of the White Noise Distribution Theory to supply the appropriate mathematical machinery to make the algorithm rigorous. In the process we construct a Gaussian measure for affine subspaces of a Hilbert space and develop a decomposition theorem for these measures.References
- Jeremy J. Becnel and Ambar N. Sengupta, An infinite-dimensional integral identity for the Segal-Bargmann transform, Proc. Amer. Math. Soc. 135 (2007), no. 9, 2995–3003. MR 2317978, DOI 10.1090/S0002-9939-07-08995-2
- Jeremy J. Becnel, Delta function for an affine subspace, Taiwanese J. Math. 12 (2008), no. 9, 2269–2294. MR 2479055, DOI 10.11650/twjm/1500405179
- Jeremy J. Becnel and Ambar N. Sengupta, The Schwartz space: Tools for white noise analysis, preprint at http://www.math.lsu.edu/ preprint/, December 2004.
- I. M. Gel′fand and N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Applications of harmonic analysis; Translated from the Russian by Amiel Feinstein. MR 0435834
- Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
- Izumi Kubo and Yoshitaka Yokoi, A remark on the space of testing random variables in the white noise calculus, Nagoya Math. J. 115 (1989), 139–149. MR 1018088, DOI 10.1017/S0027763000001586
- Hui-Hsiung Kuo, White noise distribution theory, Probability and Stochastics Series, CRC Press, Boca Raton, FL, 1996. MR 1387829
- Hui Hsiung Kuo, Gaussian measures in Banach spaces, Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin-New York, 1975. MR 0461643
- Samuel J. Lomonaco and Louis H. Kauffman, Continuous quantum hidden subgroup algorithms, Proceedings of SPIE, vol. 5105, 2003, pp. 80–88.
- Nobuaki Obata, White noise calculus and Fock space, Lecture Notes in Mathematics, vol. 1577, Springer-Verlag, Berlin, 1994. MR 1301775, DOI 10.1007/BFb0073952
Additional Information
- Jeremy J. Becnel
- Affiliation: Department of Mathematics and Statistics, Stephen F. Austin State University, Nacogdoches, Texas 75962-3040
- Email: becneljj@sfasu.edu
- Received by editor(s): July 21, 2010
- Published electronically: April 30, 2012
- Additional Notes: This research was supported by Stephen F. Austin Faculty Research Grant
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 5035-5055
- MSC (2010): Primary 60H40; Secondary 81P68
- DOI: https://doi.org/10.1090/S0002-9947-2012-05661-3
- MathSciNet review: 2931321