BIVARIATE REAL-VALUED ORTHOGONAL PERIODIC WAVELETS

In this paper, we construct a kind of bivariate real-valued orthogonal periodic wavelets. The corre-sponding decomposition and reconstruction algorithms involve only 8 terms respectively which are very simple in practical computation. Moreover, the relation between periodic wavelets and Fourier series is also discussed.

Bivariate Real-valued Box-spline Orthogonal Periodic Wavelets

In this paper, we construct a kind of bivariate real-valued orthogonal periodic box-spline wavelets. There are only 4 terms in the two-scale dilation equations. This implies that the corresponding decomposition and reconstruction algorithms involve only 4 terms respectively which are simple in practical computation. The relation between the periodic wavelets and Fourier series is also discussed.

REAL-VALUED PERIODIC WAVELETS: CONSTRUCTION ANDRELATION WITH FOURIER SERIES

In this paper, we construct the real-valued periodic orthogonal wavelets. The method presented here is new. The decomposition and reconstruction formulas involve only 4 terms respectively. It demonstrates that the formulas are simpler than that in other kinds of periodic wavelets. Our wavelets are useful in applications since it is real valued. The relation between the periodic wavelets and the Fourier series is also discussed.

Periodicities in the emplacement of large igneous provinces through the Phanerozoic:Relations to ocean chemistry and marine biodiversity evolution

Large igneous provinces （LIPs） are considered a relevant cause for mass extinctions of marine life throughout Earth＇s history. Their flood basalts and associated intrusions can cause significant release of SO4 and CO2 and consequently, cause major environmental disruptions. Here, we reconstruct the long-term periodic pattern of LIP emplacement and its impact on ocean chemistry and biodiversity from δ34Ssulfate of the last 520 Ma under particular consideration of the preservation limits of LIP records. A combination of cross-wavelet and other time-series analysis methods has been applied to quantify a potential chain of linkage between LIP emplacement periodicity, geochemical changes and the Phanerozoic marine genera record. We suggest a mantle plume cyclicity represented by LIP volumes （V） of V= （350-770） × 103km3sin（27πt/ 170 Ma）＋ （300-650）× 103 km3 sin（2πt/64.5 Ma ＋ 2.3） for t= time in Ma. A shift from the 64.5 Ma to a weaker -28-35 Ma LIP cyclicity during the Jurassic contributes together with probably independent changes in the marine sulfur cycle to less ocean anoxia, and a general stabilization of ocean chemistry and increasing marine biodiversity throughout the last -135 Ma. The LIP cycle pattern is coherent with marine biodiversity fluctuations corresponding to a reduction of marine biodiversity of -120 genera/Ma at 600 x 103 km3 LIP eruption volume. The 62-65 Ma LIP cycle pattern as well as excursion in -34Ssulfate and marine genera reduction suggest a not-vet identified found LIP event at - 440-450 Ma.

Construction of periodic wavelet frames with dilation matrix

An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.

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.