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.

翻译：非等距节点上B样条小波基的构造被推广到由任意光滑函数组合而成的分段小波。由此扩展的小波族提供了具有尽可能紧支撑且属于用户可控光滑度类的多分辨率基函数。构造过程分两个阶段进行：第一阶段，利用一组光滑函数焊接紧支撑分段光滑基函数。这些分段光滑基函数具有可细化性，即它们可以表示为在更细节点网格上构造的类似基函数的线性组合。两个尺度上的基函数之间的线性组合表达式称为细化方程或双尺度方程。第二阶段，可细化性使得能够构造小波变换。为此，将分段光滑尺度函数的细化方程分解为提升方案，进而可向后续小波基添加所需性质。除构造细节外，本文还讨论了该构造符合经典多分辨率分析框架的条件。