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

We show that if $\{f_n(t)\}_{n=-m+2}^{\infty }$ is an orthonormal system in $L^2[0,1]$ 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 $\varepsilon >0$ the function $f(t)$ can be changed on a set of measure less than $\varepsilon $ so that the corrected function has a uniformly absolutely convergent Fourier series with respect to this system.