  We exhibit the necessary range for which functions in the Sobolev spaces
$L^s_p$ can be represented as an unconditional sum of orthonormal spline
wavelet systems, such as the Battle-Lemari\'e wavelets. We also consider the
natural extensions to Triebel-Lizorkin spaces. This builds upon, and is a
generalization of, previous work of Seeger and Ullrich, where analogous results
were established for the Haar wavelet system.
