Characterization Of Matrix-dilation Filters In L~2(R~2)

Let A be a 2 x 2 integer expand matrix (That is to say, its eigenvalue's modulus is more than 1 and its entries are integers) and its determinant's absolute value is 2. Let m(s) be a 2πZ² periodic function and | m(s)|² + |m(s + 2πh0|² = 1, where h0∈Z²\A^TZ², Z denotes the integers set. m(s) is called an A-dilation generalized filterln this paper, we character all these nonseparable matrices-dilation filters in L²(R²). There are five parts in this paper. The elementary on Multi-resolution analysis in two-dimensional space is in chapter three.In the forth chapter, we mainly discuss the A-dilation filters. We discuss some properties of an A -dilation filter and its support. At the end of this chapter we present A-dilation low pass filter multipliers, and prove that a A-dilation generalized filters set is path-connected in the norm of L² ((-π,π)²). In the fifth chapter, we mainly character the A-dilation low pass filters, and at the end of this paper we give the most important conclusion: equivalent conditions of A-dilation filters being A-dilation low pass filters.