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The periodic multiresolution analysis (PMRA) and the periodic frame multiresolutionanalysis (PFMRA) provide a general recipe for the construction of periodic wavelets and periodicwavelet frames,respectively.This paper addresses PFMRAs by the introduction of the notion of spec-trum sequence.In terms of spectrum sequences,the scaling function sequences generating a normalizedPFMRA are characterized;a characterization of the spectrum sequences of PFMRAs is obtained,whichprovides a method to construct PFMRAs since its proof is constructive;a necessary and sufficient con-dition for a PFMRA to admit a single wavelet frame sequence is obtained;a necessary and sufficientcondition for a PFMRA to be contained in a given PMRA is also obtained.What is more,it is provedthat an arbitrary PFMRA must be contained in some PMRA.In the meanwhile,some examples areprovided to illustrate the general theory.
The periodic multiresolution analysis (PMRA) and the periodic frame multiresolution analysis (PFMRA) provide a general recipe for the construction of periodic wavelets and periodic wavelets frames,. This paper addresses PFMRAs by the introduction of the notion of spec-trum sequence. Terms of of spectrum sequences, the scaling function sequences generating a normalized PFMRA are characterized; a characterization of the spectrum sequences of PFMRAs is obtained, whichprovides a method to construct PFMRAs since its proof is constructive; a necessary and sufficient condition for a PFMRA to admit a single 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 areprovided to illustrate the general theory.