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 N-m. 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 N-m, it turns out that symmetry can be achieved for even in and antisymmetry for odd in, that minimum support and shift-invariance can be attained by considering the frame generators with two-scale symbols 2(-m) (1 - z)(m) and 2(-m) z(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 Lipalpha for some alpha > 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 factors, multi-wavelets, and certainly biframes (i.e., allowing the dual frames to be associated with a different refinable function). (C) 2002 Elsevier Science (USA). All rights reserved.