Arbitrarily smooth orthogonal nonseparable wavelets in R2

For each r is an element of 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-2 = +/-2I, e. g., M = ((0)(1) (2)(0)) or M = ((1)(1) (1)(-1)), then the smoothness of the wavelets improves asymptotically by 1 ? 1/2 log(2) 3 approximate to 0.2075 when r is incremented by 1. Hence they can be made arbitrarily smooth by choosing r large enough.