Arbitrarily smooth orthogonal nonseparable wavelets in ℝ2

Abstract

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 M² = ±2I, e.g., M = (^{0 2}<inf>1 0</inf>) or M = (^{1 1}<inf>1 -1</inf>), then the smoothness of the wavelets improves asymptotically by 1 - 1/2 log<inf>2</inf> 3 ≈ 0.2075 when r is incremented by 1. Hence they can be made arbitrarily smooth by choosing r large enough.