Construction of nonseparable orthonormal compactly supported wavelet bases for L 2(a"e d )

Suppose M and N are two r x r and s x s dilation matrices, respectively. Let I" (M) and I" (N) represent the complete sets of representatives of distinct cosets of the quotient groups M (-T) a"currency sign (r) /a"currency sign (r) and N (-T)a"currency sign (s) /a"currency sign (s) , respectively. Two methods for constructing nonseparable Omega-filter banks from M-filter banks and N-filter banks are presented, where Omega is a (r + s) x (r + s) dilation matrix such that one of its complete sets of representatives of distinct cosets of the quotient groups Omega (-T) a"currency sign (r+s) /a"currency sign (r+s) are I"(Omega) = {[gamma (h) (T) , zeta (q) (T) ] (T) : gamma (h) a I" (M) , zeta (q) a I" (N) }. Specially, Omega can be , where I similar to is a r x s integer matrix with M I-1 similar to being also an integer matrix. Moreover, if the constructed filter bank satisfies Lawton's condition, which can be easy to verify, then it generates an orthonormal nonseparable Omega-wavelet basis for L (2)(a"e (r+s) ). Properties, including Lawton's condition, vanishing moments and regularity of the new Omega-filter banks or new Omega-wavelet basis are discussed then. Finally, a class of nonseparable Omega-wavelet basis for L (2)(a"e (r+1)) are constructed and three other examples are given to illustrate the results. In particular, when M = N = 2, all results obtained in this paper appeared in [1].