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 -