Compactly supported tight frames associated with refinable functions

It is well known that in applied and computational mathematics, cardinal B- splines play an important role in geometric modeling (in computer-aided geo metric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth. More recently , in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of scaling functions that generate multiresolution anal yses of L-2(-infinity, infinity). However, although cardinal B-splines have compact support, their corresponding orthonormal wavelets (of Battle and L emarie) have infinite duration. To preserve such properties as self-duality while requiring compact support, the notion of tight frames is probably th e only replacement of that of orthonormal wavelets. In this paper, we study compactly supported tight frames Psi = {psi(1),..., psi(N)} for L-2(-infin ity, infinity) that correspond to some refinable functions with compact sup port, give a precise existence criterion of Psi in terms of an inequality c ondition on the Laurent polynomial symbols of the refinable functions, show that this condition is not always satisfied (implying the nonexistence of tight frames via the matrix extension approach), and give a constructive pr oof that when Psi does exist, two functions with compact support are suffic ient to constitute Psi, while three guarantee symmetry/anti-symmetry, when the given refinable function is symmetric. (C) 2000 Academic Press.