A fundamental aspect of our formulation is that the whitening operator L is naturally tied to some underlying B-spline function, which will play a crucial role in what follows. The spline connection also provides a strong link with wavelets [UB03].

In this chapter, we review the foundations of spline theory and show how one can construct B-spline basis functions and wavelets that are tied to some specific operator L. The chapter starts with a gentle introduction to wavelets that exploits the analogy with Lego blocks. This naturally leads to the formulation of a multiresolution analysis of L2(ℝ) using piecewise-constant functions and a de visu identification of Haar wavelets. We then proceed in Section 6.2 with a formal definition of our generalized brand of splines – the cardinal L-splines – followed by a detailed discussion of the fundamental notion of the Riesz basis. In Section 6.3, we systematically cover the first-order operators with the construction of exponential B-splines and wavelets, which have the convenient property of being orthogonal. We then address the theory in its full generality and present two generic methods for constructing B-spline basis functions (Section 6.4) and semi-orthogonal wavelets (Section 6.5). The pleasing aspect is that these results apply to the whole class of shift-invariant differential operators L whose null space is finite-dimensional (possibly trivial), which are precisely those that can be safely inverted to specify sparse stochastic processes.