Generalized Spline Wavelets

l j l ZZ r g Then r are called orthogonal wavelets of multiplicity r if B forms an orthonormal basis of L IR We say that r are wavelets prewavelets of multiplicity r if B forms a Riesz basis of L IR and j l is orthogonal to k n f r g l n j k ZZ with j k The general theory of wavelets of multiplicity r is treated in As usual the method is based on a generalization of the notion of multiresolution analysis as introduced by Mallat and Meyer In it is shown that any basis of orthogonal wavelets with multiplicity r composed with rapidly decaying wavelets is provided by such a generalized multiresolution of multiplicity r For applications of multiwavelets for sparse representation of smooth linear operators we refer to In this paper the ideas will be used to construct spline wavelets with knots of multiplicity r In the following let m IN and r m be given integers We consider equidistant knots of multiplicity r