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Theory of Probability and Mathematical Statistics

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Wavelet orthogonal approximation of fractional generalized random fields on bounded domains

Authors: J. M. Angulo, M. D. Ruiz-Medina and V. V. Anh
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 73 (2005).
Journal: Theor. Probability and Math. Statist. 73 (2006), 1-17
MSC (2000): Primary 60G20, 60G60; Secondary 60G12, 60H40
Published electronically: January 17, 2007
MathSciNet review: 2213332
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Abstract: We consider a class of generalized random fields defined on bounded domains, which admit a white-noise linear filter representation in terms of linear operators of fractional order. We obtain two-sided estimates of the eigenvalues defining the pure point spectra of the associated class of covariance operators. We next derive an orthonormal basis of the reproducing kernel Hilbert space from an orthonormal basis of wavelet functions. An alternative orthogonal expansion to the Karhunen-Loève expansion is then obtained in terms of wavelet functions. The results derived can be applied to compute, in particular, the mean-square solution to fractional-order integro-differential equations, and to approximate least-squares linear estimates for the class of random fields considered.

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Additional Information

J. M. Angulo
Affiliation: Department of Statistics and Operations Research, University of Granada, Campus Fuente Nueva S/N, E-18071, Granada, Spain

M. D. Ruiz-Medina
Affiliation: Department of Statistics and Operations Research, University of Granada, Campus Fuente Nueva S/N, E-18071, Granada, Spain

V. V. Anh
Affiliation: School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Q 4001, Australia

Keywords: Fractional generalized random field, fractional stochastic partial differential equation, Karhunen--Lo\`{e}ve expansion, multiresolution approximation, wavelet orthogonal approximation
Received by editor(s): May 14, 2003
Published electronically: January 17, 2007
Additional Notes: Partially supported by Projects PB96-1440 and BFM2000-1465 of the DGI, Spain, and the Australian Research Council grant A10024117
Article copyright: © Copyright 2007 American Mathematical Society

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