Interpolatory quincunx quasi-tight and tight framelets (Q6628876)

In this paper, the author studies interpolatory quincunx quasi-tight and tight framelets. Using the difference-of-squares decomposition of Hermitian trigonometric polynomials, the author proves that it is always possible to construct an interpolatory quincunx quasi-tight framelet with three generators/high-pass filters from any given interpolatory quincunx refinable function or refinement filter. He provides a new way to construct interpolatory quincunx quasi-tight framelets with high-order vanishing moments from an arbitrary interpolatory quincunx refinement filter. Also, the author proves that if the interpolatory refinement filter satisfies the sum-of-squares condition, one can always construct an interpolatory quincunx tight framelet with high-order vanishing moments. Furthermore, if the refinement filter has symmetry, the author provides a technique to impose symmetry constraints when performing the sum-of-squares decomposition so that the high-pass filters can also have symmetry. Finally, the author provides several examples to illustrate the main results.