Sparse Representation of Random Field Via Probability Chaos

ABSTRACT The main purpose of the article is to develop adaptive sparse representation theory by functions of a fixed random variable. The results are divided into two parts. One is a generalization of completeness of orthogonal polynomials in all cases where the set of monomials functions, , is contained in for a PDF supported in a compact interval . What we offer is unified treatment for all PDFs having compact support that include, and extend the classical finite interval case to its widest possibility. The second part introduces a new sparse representation methodology for random variables of second order contained in the Bochner type spaces with systems of functions of any fixed random variable of second moment.