A Study Of Approximation Order And Balancing Property Of Multiwavelet

Abstract: A multiwavelet is a wavelet generated by a scaling vector function that consists of several functions. Muliwavelet is associated with multiresolution analysis (MRA^r). A vector function Φ(t) = [φ₀,(t),φ₁(t),...,φ_r-1(t)]^T is called a multiscaling function of multiplicity r if the functions φ_i(t)∈L¹(R), (i = 0,...,r-1) generate MRA^r. More precisely, an MRA of multiplicity r is a nested sequence of closedsubspaces V_j in L²(R) satisfying conditions:5) There exist functions such that they forms a Riesz basis ofwhereLet W₀ is the orthogonal complement of V₀ in V_-1 If the integer translates of a set offunctions form an orthonormal basis of W₀, thenare called a set of orthogonal multiwavelets. The muliscaling functions of a MRA^r satisfy a matrix refinable equation:and the multiwavelets satisfy:Multiscaling functions and multiwavelets naturally generalize the scalar scaling functions and scalar wavelets. They can possess all properties simultaneously such as short support, orthogonality, symmetry and vanishing moments or higher approximation order which are superior to scalar wavelet. This is the reason why research and application of multiwavelets are more and more important in science, technology and engineering area.After a simple review of scalar wavelet and multiwavelet, this paper first introduces the two-scale similarity transform (TST) of V. Strela which is used to increase the approximation order of multiscaling function. Based on this method, different transformmatrices M_r(ω) s are constructed to increase the approximation order of finite element multiscaling function and GHM scaling function, which can reach any given integer. At the same time, short support and symmetry are preserved. Owing to the defect of TST that it cannot keep orthogonality, a family of functions that are biorthogonal to the GHM-based scaling functions and possessing some approximation order are found, which make the new families of muliscaling functions more valuable in application.Next, balancing property of multiwavelet is investigated. There is a natural relation of approximation and balancing, but the doubly infinite block matrices associated to them are different (the arrangement of column are reverse). This is disadvantageous to investigate the inner relation of them. For convenience, the paper associates the approximation of multiwavelet with the doubly infinite block matrix defining balancing, and based of this matrix gains a sufficient and necessary condition of a multiscaling function have approximation order p . Further, a theorem that is prone to estimate the approximation order of a multiscaling function is proved. By the help of these theorem, a fairly simple proof is given to deduce that a multiwavelet system with balancing order p must have a multiscaling function with approximation order p . After introduce the special interpolating condition of multiwavelet, using same theorems, there is the conclusion that if a multiscaling function have approximation order p and satisfy interpolating condition, the system is p order balanced.At last, the Coifman condition in scalar wavelet is generated to multiwavelet. Defining the discrete Coifman condition of multiplicity 2, the paper proves that the Coifman condition of multiwavelet of multiplicity 2 is equivalent to the discrete Coifman condition. The conclusion that if a multiscaling function have approximation order p and satisfy p -order Coifman condition simultaneously, the multiwavelet system has balancing order p, is deduced in the end of this paper.