Introduction

Another method to construct biorthogonal wavelets involves convolution and therefore, called convolution method. We shall discuss the convolution method in this paper. The convolution method appeared in Daubechies [21]. Later on, it was generalized to any dilation factor d in the univariate case in [61]. The convolution method was generalized to any dimension and any di­ lation matrix by Han in [31, 33] to construct both biorthogonal wavelets and dual wavelet frames. Recently, the convolution method was systematically studied by Ji, Riemenschneider and Shen [40] where the asymptotic smooth­ ness analysis of the dual scaling functions constructed by the convolution method was provided. Some other constructions of biorthogonal wavelets in the literature are also convolution based or related, for example, [38, 56].