Narrow Results

When constructing multivariate framelets, it is often unavoidable to work with matrices of multivariate trigonometric polynomials and complicated matrix decomposition problems. These problems become even harder when good properties such as high-order vanishing moments are required on the framelets. In this paper, we establish a new method for constructing multivariate dual framelets with high-order vanishing moments. The underlying scheme of our algorithm is the famous Mixed Extension Principle that allows us to derive the high-pass filters (or framelet generators) from a given pair of refinable filters with high-order linear-phase moments. Our method only involves two steps: (1) directly constructing the first few pairs of high-pass filters by using the linear-phase moment conditions of the refinement filters; (2) solving a system of linear equations to obtain the rest of the high-pass filters. Both are easy to implement for scientific computation, regardless of what dimension or dilation matrix we work with. Apart from high-order vanishing moments, we will see that if the refinement filters take coefficients from some subfield $𝔽$ of $ℂ$ that is closed under complex conjugation, so do the high-pass filters. Furthermore, our algorithm gives the upper bounds for the number of high-pass filters in arbitrary dimensions. At the end of the paper, we will give several illustrative examples, from which we can also see that the support sizes of the high-pass filters are comparable with those of the refinement filters.