Gmra-Based Construction of Framelets in Reducing Subspaces of L2(ℝd)

This paper develops GMRA-based construction procedures of Parseval framelets in the setting of reducing subspaces of L2(ℝd). A unitary extension principle is established; in particular, for a general expansive matrix A with |det A| = 2, an explicit construction of Parseval framelets is obtained. Some examples are also provided to illustrate the generality of our theory.

[1]  Zuowei Shen.  Affine systems in L 2 ( IR d ) : the analysis of the analysis operator , 1995 .

[2]  W. Lawton.  Tight frames of compactly supported affine wavelets , 1990 .

[3]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[4]  R. DeVore,et al.  On the construction of multivariate (pre)wavelets , 1993 .

[5]  X. Dai,et al.  Subspaces with normalized tight frame wavelets in R , 2001 .

[6]  A. Ron,et al.  Affine Systems inL2(Rd): The Analysis of the Analysis Operator , 1997 .

[7]  Frame wavelets in subspaces of ²(ℝ^{}) , 2002 .

[8]  X. Dai,et al.  Frame wavelets in subspaces of L^2(R^d) , 2002 .

[9]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[10]  Marcin Bownik,et al.  Construction and reconstruction of tight framelets and wavelets via matrix mask functions , 2009 .

[11]  A. Ron,et al.  Tight compactly supported wavelet frames of arbitrarily high smoothness , 1998 .

[12]  R. DeVore,et al.  The Structure of Finitely Generated Shift-Invariant Spaces in , 1992 .

[13]  H. Volkmer.  FRAMES OF WAVELETS IN HARDY SPACE , 1995 .

[14]  Marcin Bownik,et al.  Tight frames of multidimensional wavelets , 1997 .

[15]  I. Daubechies,et al.  Framelets: MRA-based constructions of wavelet frames☆☆☆ , 2003 .

[16]  Qiaofang Lian,et al.  Reducing subspace frame multiresolution analysis and frame wavelets , 2007 .

[17]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[18]  K. Seip.  Regular sets of sampling and interpolation for weighted Bergman spaces , 1993 .

[19]  Marcin Bownik.  Intersection of dilates of shift-invariant spaces , 2008 .

[20]  C. Chui,et al.  Compactly supported tight frames associated with refinable functions , 2000 .

[21]  X. Dai,et al.  The existence of subspace wavelet sets , 2003 .

[22]  Yun-Zhang Li,et al.  Density results for Gabor systems associated with periodic subsets of the real line , 2009, J. Approx. Theory.

[23]  B. Han.  On Dual Wavelet Tight Frames , 1997 .

[24]  A. Petukhov.  Explicit Construction of Framelets , 2001 .