Wavelet orthogonal approximation of fractional generalized random fields on bounded domains. (English) Zbl 1118.60043

Angulo, J. M.; Ruiz-Medina, M. D.; Anh, V. V.

Teor. Jmovirn. Mat. Stat. 73, 1-16 (2005) and Theory Probab. Math. Stat., Vol 73, 1-17 (2006).

The authors deal with generalized random fields defined on fractional Sobolev spaces on bounded domains characterized by their fractional-order pure point spectra. These fields are referred to as fractional generalized random fields (FGRFs). It is assumed that the norm of the associated reproducing kernel Hilbert space (RKHS) is equivalent to the norm defined on a fractional Sobolev space of certain order. This condition is equivalent to the existence of a dual generalized random field, and to the existence of a white-noise linear filter representation. It is proved that the spectral properties of the corresponding class of covariance operators are equivalent to those presented by compact embeddings between fractional Sobolev spaces on bounded domains. The orthonormal bases constructed for the RKHSs of a FGRF and its dual provide a non-redundant description of the second-order structure of the FGRF. Such bases then allow the definition of an orthogonal expansion in terms of wavelets for the FGRFs. This type of expansion provides an alternative to the Karhunen-Loève expansion useful in the cases where the second-kind Fredholm integral equation defining the covariance eigenfunction system is not explicitly solvable.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60G60	Random fields
60G20	Generalized stochastic processes
60G12	General second-order stochastic processes
60H40	White noise theory
46F25	Distributions on infinite-dimensional spaces
46N30	Applications of functional analysis in probability theory and statistics

Keywords:

fractional generalized random field; fractional stochastic partial differential equation; Karhunen–Loève expansion; multiresolution approximation; wavelet orthogonal approximation