Interpolatory dual framelets with a general dilation matrix

Interpolatory filters are of great interest in subdivision schemes and wavelet analysis. Due to the high-order linear-phase moment property, interpolatory refinement filters are often used to construct wavelets and framelets with high-order vanishing moments. In this paper, given a general dilation matrix [Formula: see text], we propose a method that allows us to construct a dual [Formula: see text]-framelet from an arbitrary pair of [Formula: see text]-interpolatory filters such that all framelet generators/high-pass filters (1) have the interpolatory properties; (2) have high-order vanishing moments. Our method is easy to implement, as the high-pass filters are either given in explicit formulas or can be obtained by solving specific linear systems. Motivated by constructing interpolatory dual framelets, we can further deduce a method to construct an interpolatory quasi-tight framelet from an arbitrary interpolatory filter. If, in addition, the refinement filters have symmetry, we will perform a detailed analysis of the symmetry properties that the high-pass filters can achieve. We will present several examples to demonstrate our theoretical results.