On the Representation of Measurable Functions by Absolutely Convergent Orthogonal Spline Series

Abstract

Abstract: We show that if {fn(t)}∞n=−m+2 is an orthonormal system in L2[0,1] consisting of splines of order m with dyadic rational knots and f(t) is an a.e. finite measurable function, then, first, there exists a series with respect to this system that converges absolutely a.e. to this function and, second, for any Ɛ>0 the function f(t) can be changed on a set of measure less than Ɛ so that the corrected function has a uniformly absolutely convergent Fourier series with respect to this system.