Interpolatory quincunx quasi-tight and tight framelets

Constructing multivariate tight framelets is a challenging problem in wavelet and framelet theory. The problem is intrinsically related to the Hermitian sum of squares decomposition of multivariate trigonometric polynomials and the spectral factorization of multivariate trigonometric polynomial matrices. To circumvent the relevant difficulties, the notion of a quasi-tight framelet has been introduced in recent years, which generalizes the concept of tight framelets. On one hand, quasi-tight framelets behave similarly to tight framelets. On the other hand, compared to tight framelets, quasi-tight framelets have much more flexibility and advantages. Motivated by several recent studies of multivariate quasi-tight and tight framelets, we work on quincunx quasi-tight and tight framelets with the interpolatory properties in this paper. We first show that from any interpolatory quincunx refinement filter, one can always construct an interpolatory quasi-tight framelet with three generators. Next, we shall present a way to construct interpolatory quincunx quasi-tight framelets with high-order vanishing moments. Finally, we will establish an algorithm to construct interpolatory quincunx tight framelets from any interpolatory quincunx refinement filter that satisfies the so-called sum-of-squares (SOS) condition. All our proofs are constructive, and several examples in dimension \(d=2\) will be provided to illustrate our main results.