Generalized free Gaussian white noise

Hakeem Othman

doi:10.14419/ijams.v4i1.5911

Authors

Hakeem Othman Head of Department of Mathematics, University College of Al-Qunfudah,Umm Alqura Uniersity, Alkhalidya, Alqunfuhah city, Kingdom of Saudi

DOI:

https://doi.org/10.14419/ijams.v4i1.5911

Published:

2016-03-27

Keywords:

Chebychev polynomials, Wigner semicircle distribution, Fourier transform, Wigner semicircle white noise

Abstract

Based on an adequate new Gel'fand triple, we construct the infinite dimensional free Gaussian white noise measure \(\mu\) using the Bochner-Minlos theorem. Next, we give the chaos decomposition of an \(L^{2}\) space with respect to the measure \(\mu\).

References

[1] N. Asai, I. Kubo and H.-H. Kuo, Multiplication Renormalization and Generating Function I., Taiwanese Journal of Mathematics, Vol. 8, No. 4 (2004), 583-628.

[2] T.S. Chihara, â€An introduction to Orthogonal Polynomializationâ€ ,Gordon and Breach, New York, 1978.

[3] G. Gasper and M. Rahman, â€ Basic hypergeometric seriesâ€, Vol 35 of Encyclopedia Of Mathematics And Its Application, Cambridge Universty Press, Cambridge (1990).

[4] Yu. G. Kondratiev, J.L. Silva, L. Streit and G.F. Us, â€Analysis on Pois-son and Gamma spaceâ€, Infinite dimensional anal. Quant. Probab, Vol. 1 No. 1 (1998), 91-118.

[5] H. Van Leeuwen and H. Maassen, â€A q-deformation of the Gauss distributionâ€, Journal of mathematical physic, 36(9), 4743-4756 [1995].

[6] H. Rguigui, â€Quantum l -potentials associated to quantum Ornstein-Uhlenbeck semigroupsâ€, Chaos, Solitons & Fractals, Volume 73, April 2015, Pages 80-89.