Splinets – efficient orthonormalization of the B-splines

The problem of orthogonalization of the B-spline basis is discussed for both equally and arbitrarily spaced knots. A new efficient orthogonalization is proposed and contrasted with some previous methods. This new orthogonal basis of the splines is better visualized as a net of orthogonalized functions rather than a sequence of them. The net is spread over different support rangeution and different locations resembling in this respect wavelets bases. For this reason the constructed basis is referred to as a splinet and features some clear advantages over other spline bases. The splinets exploit nearly orthogonalization featured by the B-splines themselves and through this gains are achieved at two levels: a locality that is exhibited through small size of the total support of a splinet and computational efficiency that follows from a small number of orthogonalization procedures needed to be performed on the B-splines to reach orthogonality. The original not orthogonalized B-splines have the total support on which they jointly sit of the order O(1) relatively to the length of the underlying common interval on which they are constructed. On the other hand, the orthogonalized bases previously discussed in the literature have the total support of the order O(n), where n stands for the number of knots which is also the number of basis functions (up to a constant). The size of the total support for a splinet is of the order of O(log n) which is hypothesized to be optimal among the orthogonal spline bases. Moreover, computational costs, of the proposed orthogonalization has the same computational cost as the standard methods. A dedicated representation of splines is proposed and implemented into the numerical package and rendered as an R- package splinets.