Matrix splitting with symmetry and symmetric tight framelet filter banks with two high-pass filters

The oblique extension principle introduced in Chui et al. (2002), Daubechies et al. (2003) [3,5] is a general procedure to construct tight wavelet frames and their associated filter banks. Symmetric tight framelet filter banks with two high-pass filters have been studied in Han and Mo (2004), Mo and Zhuang (in press), Petukhov (2003) [13,16,17]. Tight framelet filter banks with or without symmetry have been constructed in many papers in the literature. This paper is largely motivated by several results in Han (2010), Han and Mo (2004), Petukhov (2003) [11,13,17] to further study tight wavelet frames and their associated filter banks with symmetry and two high-pass filters. Our study not only leads to a simpler algorithm for the construction of tight framelet filter banks with symmetry and two high-pass filters, but also allows us to obtain a wider class of tight wavelet frames with symmetry which are not available in the current literature. The key ingredient in our investigation is a complete characterization of splitting positive semi-definite 2×2 matrices of Laurent polynomials with symmetry. Several examples are provided to illustrate the results and algorithms in this paper.