Several Problems On Wavelet Bi-frame With Canonical Filters In Low Dimension And Topological Invariants Of Graphs

Due to the greater freedom and the complexity of calculation,the construction of multivariate wavelet bi-frames has always been one of the difficult problems in the study of wavelet frames.In general,the wavelets defined by the canonical filters associated with the refinement mask of B-spline or box do not form a wavelet bi-frame.As to this issue,this paper gives a method for the construction of multivariate wavelet bi-frames with canonical filters by adding filters to the original system.Specifically,for a pair of given lowpass filters and 2d—1 pairs canonical filters,we first prove that the smallest number of highpass filters which need to be added to generate a semi-canonical wavelet bi-frame filter is 2d+1—1(d = 1,2,3)by using the mixed unitary extension principle,and then we obtain the construction method based on the relationship between the filters and the polyphase matrix.For a graph,the general Zeroth-order Randic index R?0 is defined as the sum of the ?-th,?#0 and ?? 1,power of the vertex degree.Let Hn be the class of all maximal outerplanar graphs on n vertices,and Tn,k be the class of trees with n vertices of which k vertices have the maximum degree.In this paper,we determine the extremal trees of the class Hn and Tn,k i.e.,those with maximal R_?~0 or minimal R_?~0 in the two classes.The corresponding extremal graphs are also completely characterized.