TY - JOUR
T1 - Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments
AU - He, Wenjie
AU - Chui, Charles K.
AU - Stocker, Joachim
AU - Sun, Qiyu
AU - Stöckler, Joachim
N1 - 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 M≥2, the standard "matrix...
PY - 2003/2/1
Y1 - 2003/2/1
N2 - When a Cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L 2 = L 2 (IR) 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, Han, Ron, and Shen) 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 Mdilation 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
AB - When a Cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L 2 = L 2 (IR) 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, Han, Ron, and Shen) 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 Mdilation 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
UR - https://www.umsl.edu/mathcs/about/People/Faculty/WenjieHe/He%20he%20Wen/compactly.pdf
UR - https://link.springer.com/article/10.1023/A:1021318804341
U2 - 10.1023/A:1021318804341
DO - 10.1023/A:1021318804341
M3 - Article
VL - 18
JO - Advanced Computational Mathematics
JF - Advanced Computational Mathematics
ER -