Compactly supported tight affine frames with integer dilations and maximum vanishing moments

Abstract

When a cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L2 = L 2(ℝ) with dilation integer factor M ≥ 2, the standard "matrix extension" approach for constructing compactly supported tight frames has the limitation that at least one of the tight frame generators does not annihilate any polynomial except the constant. The notion of vanishing moment recovery (VMR) was introduced in our earlier work (and independently by Daubechies et al.) for dilation M = 2 to increase the order of vanishing moments. This present paper extends the tight frame results in the above mentioned papers from dilation M = 2 to arbitrary integer M ≥ 2 for any compactly supported M-dilation scaling functions. It is shown, in particular, that M compactly supported tight frame generators suffice, but not M - 1 in general. A complete characterization of the M-dilation polynomial symbol is derived for the existence of M - 1 such frame generators. Linear spline examples are given for M = 3, 4 to demonstrate our constructive approach.