International Journal of Pure and Applied Mathematics ————————————————————————– Volume 47 No. 4 2008, 459-473 STATISTICAL LEARNING METHODS FOR UNIFORM APPROXIMATION BOUNDS IN MULTIRESOLUTION SPACES Mark A. Kon 1 , Louise A. Raphael 2 § 1 Department of Mathematics and Statistics Boston University Boston, MA 02215, USA e-mail: mkon@math.bu.edu 2 Department of Mathematics Howard University Washington, DC 20059, USA e-mail: lraphael@howard.edu Abstract: New constructive and non-constructive non-asymptotic uniform error bounds for approximating functions in L 2 s (R d ),d ≥ 1, by finite compactly supported multiresolution expansions are proved using approximation theoretic bounds derived from statistical learning theory. AMS Subject Classification: 41A25, 41A65, 68T05 Key Words: statistical learning theory (SLT), VC dimension, multiresolution analysis (MRA), wavelets, reproducing kernel Hilbert space (RKHS) 1. Introduction Given a function f 0 in an L 2 Sobolev space L 2 s (R d ), s> 0,d ≥ 1, we will examine here both constructive and non-constructive weighted L ∞ approxima- tions of f 0 using finite combinations of compactly supported scaling functions or, equivalently, compactly supported wavelets. We follow an approach which parallels statistical learning theory (SLT) methods first developed by Girosi [5]. Our non-constructive result is an exten- sion of one appearing in [11]. The constructive results are probabilistic, and Received: July 24, 2008 c  2008, Academic Publications Ltd. § Correspondence author