Some bivariate smooth compactly supported tight framelets with three generators. (English) Zbl 1276.42026

San Antolín, A.; Zalik, R. A.

Abstr. Appl. Anal. 2013, Article ID 818907, 7 p. (2013).

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 [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.

Reviewer: Bin Han (Edmonton)

MSC:

42C40

Nontrigonometric harmonic analysis involving wavelets and other special systems

Keywords:

Bivariate tight framelets; dilation matrix; refinable functions; refinable functions; smooth compactly supported tight framelets

Citations:

Zbl 0886.65013

References:

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