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 RdRd. MEP and MOEP provide us with criteria for the construction of wavelet bi-frames on RdRd. 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.