Abstract. New algorithms for fast wavelet transforms with biorthogonal spline wavelets on nonuniform grids are presented. In contrary to classical wavelet transforms, the algorithms are not based on filter coefficients, but on algorithms for B-spline expansions (differentiation, Oslo algorithm, etc.). Due to inherent properties of the spline wavelets, the algorithm can be modified for spline grid refinement or coarsening. The performance of the algorithms is demonstrated by numerical tests of the adaptive spline methods in circuit simulation.Key words. Splines, spline wavelets, free knot spline approximation AMS subject classifications. 65D07, 41A15, 65T60, 42C401. Introduction. Since the dawn of wavelet theory spline wavelets have been always of particular interest. This includes orthogonal spline wavelets [3,35], semiorthogonal spline wavelets [16,17] as well as biorthogonal wavelets [19]. An exceptional property of spline wavelets is that they possess an explicit representation (in terms of piecewise polynomials), while most other wavelets of interest are only described by their two-scale relation. This permits extra flexibility, e.g. for the construction of wavelets on the interval [15,21] or the evaluation of non-linear mappings of wavelet expansions [11,18,24].The wavelet constructions above are based on equidistant spline knots. However, since a spline can be defined for any given grid, it arises the question if spline wavelet constructions on nonuniform grids are possible. There have been several publications on semi-orthogonal spline wavelets on non-uniform grids (see e.g. [14,33,37]). Although these wavelets do not have a sparse decomposition relation, there are fast algorithms, which solve a banded linear system involving the reconstruction coefficients (cf. [41]). However, in some cases finite decomposition relations as they appear for biorthogonal spline wavelets [19,21] may be of interest. A constructive proof of the existence of such spline wavelets in the nonuniform setting was given in [22], based on results of banded matrices with banded inverses. However, no algorithms based on this approach are provided.In [4] we have given sufficient and necessary conditions for the existence of finite reconstruction and decomposition relations. However, an algorithm based on this relation requires the computation of many coefficients, which is time and memory consuming. Thus, we have developed a direct approach based on known properties of spline functions. Here we will present algorithms for the fast wavelet transform for non-uniform spline wavelets, which represent a generalization of the biorthogonal spline wavelets from [19]. By a small modification we obtain also an algorithm for adaptive knot removal, which permits to reduce the size of a spline representation with the approximation error under control. Furthermore, we introduce an adaptive spline approximation method with adaptive grid refinement. These algorithms were
