Quasi-Interpolatory Refinable Functions and Contruction of Biorthogonal Wavelet Systems

We present a new family of compactly supported and symmetric biorthogonal wavelet systems. Each refinement mask in this family has tension parameter ω. When ω = 0, it becomes the minimal length biorthogonal Coifman wavelet system [22]. Choosing ω away from zero, we can get better smoothness of the refinable functions at the expense of slightly larger support. Though the construction of the new birothogonal wavelet systems, in fact, starts from a new class of quasi-interpolatory subdivision schemes, we find that the refinement masks accidently coincide with the ones by Cohen, Daubechies and Feauveau [3, §6.C] (or [8, §8.3.5]), which are designed for the purpose of generating biorthogonal wavelets close to orthonormal cases. However, the corresponding mathematical analysis is yet to be provided. In this study, we highlight the connection between the quasi-interpolatory subdivision schemes and the masks by Cohen, Daubechies and Feauveau, and then we study the fundamental properties of the new biorthogonal wavelet systems such as regularity, stability, linear independence, vanishing moments and accuracy.