Constructing wavelets by welding segments of smooth functions

The construction of B-spline wavelet bases on nonequispaced knots is extended to wavelets that are piecewise segments from any combination of smooth functions. The extended wavelet family thus provides multiresolution basis functions with support as compact as possible and belonging to a user controlled smoothness class. The construction proceeds in two phases. In the first fase, a set of smooth functions is used in the welding of compact supported, piecewise smooth basis functions. These piecewise smooth basis functions are refinable, meaning that they can be written as linear combinations of similar basis functions constructed on a fined grid of knots. The expression of the linear combination between the bases at two scales is known as a refinement or two-scale equation. In the second phase, the refinabability enables the construction of a wavelet transform. To this end, the refinement equation of the piecewise smooth scaling functions is factored into a lifting scheme, to which the desired properties of the subsequent wavelet basis can then be added. Next to the details of the construction, the paper discusses the conditions for it to fit into the classical framework of multiresolution analyses.