Arbitrarily Smooth Orthogonal Nonseparable Wavelets in $\R^2$
Eugene A. Belogay, Yang Wang · SIAM Journal on Mathematical Analysis · 1999
For each $r\in\N$, 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 $\detwo$. We prove that if $\M^2=\pm 2I$, e.\,g., %%$\M=\ourmtx$ or $\M=\qsymtx$, $M=({0\,\,2 \atop 1\,\,0})$ or $M=({1\,\,\phantom{-}1 \atop 1\,\,-1})$, then the smoothness of the wavelets improves asymptotically by $1-\frac{1}{2}\log_23 \approx 0.2075$ when r is incremented by 1. Hence they can be made arbitrarily smooth by choosing r large enough.