Compactly supported tight and sibling frames with maximum vanishing moments

The notion of vanishing-moment recovery (VMR) functions is introduced in this paper for the construction of compactly supported tight frames with two generators having the maximum order of vanishing moments as determined by the given refinable function, such as the mth order cardinal B-spline Nm. Tight frames are also extended to “sibling frames” to allow additional properties, such as symmetry (or antisymmetry), minimum support, “shift-invariance,” and inter-orthogonality. For Nm, it turns out that symmetry can be achieved for even m and antisymmetry for odd m, that minimum support and shift-invariance can be attained by considering the frame generators with twoscale symbols 2−m(1 − z)m and 2−mz(1 − z)m, and that inter-orthogonality is always achievable, but sometimes at the sacrifice of symmetry. The results in this paper are valid for all compactly supported refinable functions that are reasonably smooth, such as piecewise Lipα for some α > 0, as long as the corresponding two-scale Laurent polynomial symbols vanish at z = −1. Furthermore, the methods developed here can be extended to the more general setting, such as arbitrary integer scaling actors, multi-wavelets, and certainly biframes (i.e., allowing the dual frames to be associated with a different refinable function).