ARBITRARILY SMOOTH ORTHOGONAL NONSEPARABLEWAVELETS IN

For each r ∈ N, we construct a family of bivariate orthogonal wavelets with compact support that are nonseparable and have vanishing moments of order r or less. The starting point of the construction is a scaling function that satisfies a dilation equation with special coefficients and a special dilation matrix M : the coefficients are aligned along two adjacent rows, and |det(M)| = 2. We prove that if M = ±2I, e. g., M = ( 0 2 1 0 ) or M = ( 1 1 1 −1 ), then the smoothness of the wavelets improves asymptotically by 1 − 1 2 log2 3 ≈ 0.2075 when r is incremented by 1. Hence they can be made arbitrarily smooth by choosing r large enough.