In this work, a general theory for the construction of biorthogonal multiwavelets is developed. Starting with finitely generated shift invariant (FSI) spaces {dollar}V{dollar} and {dollar}tilde V,{dollar} with the property that the generating functions are supported on ({dollar}-{dollar}1,1) and satisfy a local linear independence condition, necessary and sufficent conditions are given such that dual bases for {dollar}V{dollar} and {dollar}tilde V{dollar} exist. When such dual FSI spaces generate multi-resolution analyses {dollar}(Vsb{lcub}p{rcub})sb{lcub}p{lcub}in{rcub}Z{rcub}{dollar} and {dollar}(tilde Vsb{lcub}p{rcub})sb{lcub}p{lcub}in{rcub}Z{rcub}{dollar} conditions related to the matrix dilation coefficients of the scaling functions are given, that lead to a construction of a family of biorthogonal multiwavelets that are continuous, compactly supported, and symmetric (antisymmetric). This construction is particularly simple and easy to implement numerically. A family of such dual multiresolution analyses are constructed, each generated by two compactly supported, continuous, and symmetric scaling functions. Once such scaling functions are constructed, a family of biorthogonal multiwavelets are constructed. Finally, such dual multi-resolution analyses {dollar}(Vsb{lcub}p{rcub})sb{lcub}p{lcub}in{rcub}Z{rcub}{dollar} and {dollar}(tilde Vsb{lcub}p{rcub})sb{lcub}p{lcub}in{rcub}Z{rcub}{dollar} generate dual filters {dollar}{lcub}Msb0(z), tilde Msb0(z){rcub}{dollar} that must satisfy the following Bezout equation:{dollar}{dollar} Msb0(z)tilde Msbsp{lcub}0{rcub}{lcub}dagger{rcub}(z)+Msb0({lcub}-{rcub}z)tilde Msbsp{lcub}0{rcub}{lcub}dagger{rcub}({lcub}-{rcub}z)=I.{dollar}{dollar}Starting with a matrix polynomial {dollar}Msb0(z){dollar} of fixed degree, all solutions {dollar}tilde Msbsp{lcub}0{rcub}{lcub}dagger{rcub}(z){dollar} to Bezout's equation are categorized.
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机译:在这项工作中,发展了构造双正交多小波的一般理论。从有限生成的移位不变(FSI)空间{dollar} V {dollar}和{dollar} tilde V开始,{dollar}具有支持生成函数的属性({dollar}-{dollar} 1,1)和如果满足局部线性独立条件,则给出必要条件和充分条件,使得存在{dollar} V {dollar}和{dollar} tilde V {dollar}的双重碱基。当此类双FSI空间生成多分辨率分析时,{rcub} Z {rcub} {dollar}和{dollar}(tilde Vsb {dollar}(Vsb {lcub} p {rcub})sb {lcub} p {lcub}给出了与缩放函数的矩阵扩张系数相关的条件,从而构造了一个双正交的族连续,紧凑支持和对称(反对称)的多小波。这种构造特别简单并且易于在数值上实现。构建了此类双重多分辨率分析的族,每个分析均由两个紧密支持的,连续且对称的缩放函数生成。一旦构造了这样的缩放函数,就构造了双正交多小波族。最后,在{rcub} Z {rcub} {dollar}和{dollar} {tilde Vsb {lcub}中,{dollar}(Vsb {lcub} p {rcub})sb {lcub} p {lcub} p {rcub})sb {lcub} p {lcub} in {rcub} Z {rcub} {dollar}生成双重过滤器{dollar} {lcub} Msb0(z),波浪线Msb0(z){rcub} {dollar}必须满足以下Bezout公式:{dollar} {dollar} Msb0(z)波浪号Msbsp {lcub} 0 {rcub} {lcub} dagger {rcub}(z)+ Msb0({lcub}-{rcub} z)波浪号Msbsp {lcub} 0 {rcub} {lcub} dagger {rcub}({lcub}-{rcub} z)= I。{dollar} {dollar}从矩阵多项式{dollar} Msb0(z){dollar}开始度,对Bezout方程的所有解{dollar} tilde Msbsp {lcub} 0 {rcub} {lcub} dagger {rcub}(z){dollar}都进行了分类。
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