The construction of multivariate periodic wavelet bi-frames

This paper addresses periodic wavelet bi-frames associated with general expansive matrices. Periodization is an important method to obtain periodic wavelets from wavelets on R^d${\mathbb{R}}^d$. MEP and MOEP provide us with criteria for the construction of wavelet bi-frames on R^d${\mathbb{R}}^d$. Based on periodization techniques, MEP and MOEP, periodic wavelet bi-frames associated with the dyadic matrix have been constructed. However, the problem of constructing periodic wavelet bi-frames associated with general expansive matrices is still open. The geometry of a general expansive matrix is much more complicated than the dyadic matrix. In this paper, with the help of quasi-norms, MEP and MOEP we construct periodic wavelet bi-frames associated with general expansive matrices.