Some bivariate smooth compactly supported tight framelets with three generators (Q370236)

For \(2\times 2\) expansive integer matrices \(A\) such that \(|\det (A)|=2\), using convolution of refinable characteristic functions in Proposition~1 and the oblique extension principle in Theorem~A, this paper constructs bivariate smooth compactly supported tight framelets with three generators. Since \(|\det(A)|=2\), the bivariate symbol \(P(t)\) is essentially one-dimensional and therefore the authors can employ the Fejér-Riesz lemma to obtain a desired trigonometric polynomial \(L\) in [\textit{T. Goodman} et al., Adv. Comput. Math. 7, No. 4, 429--454 (1997; Zbl 0886.65013)]. This allows the authors to obtain compactly supported tight framelets with three generators using a known construction method in Theorem B. The constructed tight framelets have arbitrarily high smoothness, but have only one vanishing moment.