| 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. |
