The spectrum sequences of periodic frame multiresolution analysis

The periodic multiresolution analysis (PMRA) and the periodic frame multiresolution analysis (PFMRA) provide a general recipe for the construction of periodic wavelets and periodic wavelet frames, respectively. This paper addresses PFMRAs by the introduction of the notion of spectrum sequence. In terms of spectrum sequences, the scaling function sequences generating a normalized PFMRA are characterized; a characterization of the spectrum sequences of PFMRAs is obtained, which provides a method to construct PFMRAs since its proof is constructive; a necessary and sufficient condition for a PFMRA to admit a single wavelet frame sequence is obtained; a necessary and sufficient condition for a PFMRA to be contained in a given PMRA is also obtained. What is more, it is proved that an arbitrary PFMRA must be contained in some PMRA. In the meanwhile, some examples are provided to illustrate the general theory. Supported by the National Natural Science Foundation of China (Grant No. 10671008) and the first author is also partially supported by the Excellent Talents Foundation of Beijing, China (20051D0501022), PHR(IHLB), the Project-sponsored by SRF for ROCS, SEM of China and the Scientific Research Foundation for the Excellent Returned Overseas Chinese Scholars, Beijing