Some Bivariate Smooth Compactly Supported Tight Framelets with Three Generators

Abstract

For any $2\times 2$ dilation matrix with integer entries and $|\text{det\hspace{0.17em}}A|=2$, we construct a family of smooth compactly supported tight wavelet frames with three generators in $L^2({ℝ}^2)$. Our construction involves some compactly supported refinable functions, the oblique extension principle, and a slight generalization of a theorem of Lai and Stöckler. Estimates for the degrees of smoothness are given. With the exception of a polynomial whose coefficients must in general be computed by spectral factorization, the framelets are expressed in closed form in the frequency domain, in terms of elementary transcendental functions. By means of two examples we also show that for low degrees of smoothness the use of spectral factorization may be avoided.