Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments
Source: Advances in Computational Mathematics, Volume 18, Numbers 2-4, February 2003 , pp. 159-187(29)
When a cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L2=L2(R) with dilation integer factor M2, 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 M2 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.
Document Type: Research Article
Affiliations: 1: Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, MO 63121-4499, USA, Department of Statistics, Stanford University, Stanford, CA 94305, USA 2: Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, MO 63121-4499, USA 3: Fachbereich Mathematik, Universität Dortmund, D-44221 Dortmund, Germany 4: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Publication date: February 2003