Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T02:28:16.382Z Has data issue: false hasContentIssue false
  • English
  • Français

The Wiener-Hopf integral equation for fractional Riesz-Bessel motion

Published online by Cambridge University Press:  17 February 2009

V. V. Anh ,
W. Grecksch ,
J. M. Angulo  and
M. D. Ruiz-Medina
V. V. Anh
Affiliation:
Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia. e-mail: v.anh@fsc.qut.edu.au
W. Grecksch
Affiliation:
Faculty of Mathematics and Informatics, Martin-Luther University of Halle-Wittenberg, D-06099, Halle, Germany. e-mail: grecksch@mathematik.uni-halle.d400.de
J. M. Angulo
Affiliation:
Department of Statistics and Operations Research, University of Granada, Campus Fuente Nueva S/N, E-18071, Granada, Spain. e-mail: jmangulo@goliat.ugr.es, mruiz@goliat.ugr.es
M. D. Ruiz-Medina
Affiliation:
Department of Statistics and Operations Research, University of Granada, Campus Fuente Nueva S/N, E-18071, Granada, Spain. e-mail: jmangulo@goliat.ugr.es, mruiz@goliat.ugr.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper gives an approximate solution to the Wiener-Hopf integral equation for filtering fractional Riesz-Bessel motion. This is obtained by showing that the corresponding covariance operator of the integral equation is a continuous isomorphism between appropriate fractional Sobolev spaces. The proof relies on properties of the Riesz and Bessel potentials and the theory of fractional Sobolev spaces.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 1 , July 2000 , pp. 41 - 54
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Anh, V. V. and Lunney, K. E., “Parameter estimation of random fields with long-range dependence”, Math. Comp. Modelling 21 (1995) 67–77.CrossRefGoogle Scholar
[2]Beran, J., “Statistical models for data with long-range dependence”, Statist. Science 7 (1992) 404–427.Google Scholar
[3]Dautray, R. and Lions, J. L., Mathematical analysis and numerical methods for science and technology. Vol. 2 Functional and variational methods (Springer, New-York, 1985).Google Scholar
[4]Donoghue, W. F., Distributions and Fourier Transforms (Academic Press, New York, 1969).Google Scholar
[5]Edwards, R. E., Functional Analysis (Holt, Rinehart and Winston, New York, 1965).Google Scholar
[6]Gel'fand, I. M. and Vilenkin, N. Y., Generalized Functions, Volume 4 (Academic Press, New York, 1964).Google Scholar
[7]Hutson, V. and Pym, J. S., Applications of Functional Analysis and Operator Theory (Academic Press, London, 1980).Google Scholar
[8]Liu, F. and Anh, V. V., “Numerical method for one-dimensional Wiener-Hopf integral equations of the first kind”, QUT Working Paper in Mathematics, 3/96.Google Scholar
[9]Mandelbrot, B. B. and Van Ness, J. M., “Fractional Brownian motion, fractional noises and applications”, SIAM Review 10 (1968) 422–437.CrossRefGoogle Scholar
[10]Peters, E., Fractal Market Analysis (Wiley, New York, 1994).Google Scholar
[11]Ramm, A. G., Theory and Applications of some New Classes of Integral Equations (Springer, New York, 1980).CrossRefGoogle Scholar
[12]Ramm, A. G., Random Fields Estimation Theory (Longman, Essex, 1990).CrossRefGoogle Scholar
[13]Ruiz-Medina, M. D., Angulo, J. M. and Anh, V. V., “Fractional generalised random fields”, (submitted).Google Scholar
[14]Stein, E. M., Singular Integrals and Differential Properties of Functions (Princeton University Press, New Jersey, 1970).Google Scholar
[15]Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (North-Holland, New York, 1978).Google Scholar