Construction of Refinement Masks Satisfying the Sum Rules of Order One Using Spectral Data

Abstract

A wavelet is a single function and#968;(x), whose integer translates of dyadic dilates newlineforms a basis of the space L2(R). Construction of a wavelet typically starts with finding newlinea solution to the scaling equation and#966;(x) = newlineand#8730; newline2 and#931; newlinekand#8712;Z newlinehkand#966;(2xand#8722;k). The solution of a scaling newlineequation is known as the scaling function. A wavelet is constructed using the notion newlineof a multiresolution analysis, which is a collection of closed subspaces . . . and#8834; Vand#8722;1 and#8834; newlineV0 and#8834; V1 and#8834; V2 and#8834; . . . of L2(R), satisfying certain properties. The space V0 is generated newlineby a scaling function and#966;(x). One variant of the wavelets is the multiwavelets, where newlineinteger translates of dyadic dilates of more than one function forms a basis of L2(R). newlineIn this case, the subspace V0 in a multiresolution analysis is generated by more than newlineone function. h Instead of a scaling function, we have a vector-valued function and#934;(x) = newlineand#966;1(x) and#966;2(x) . . . and#966;n(x) newlineiT newline. This vector valued function, and#934;, is called a refinable newlinefunction vector. The significance of multiwavelets lies in the fact that the simultaneous newlineinclusion of more properties is possible. newlineIn the case of multiwavelets, the scaling equation is a matrix equation known newlineas the matrix refinement equation. The matrix refinement equation is of the form newlineand#934;(x) = newlineand#8730; newline2 newlineland#931; newlinek=0 newlineHkand#934;(2xand#8722;k), Hk and#8712; Cn×n. One factor which contains the coefficient matrices newlineHks is the refinement mask given by, H(and#958; ) = newline1 newlineand#8730; newline2 newlineland#931; k=0 newlineHkeand#8722;iand#958; k. The existence and newlineproperties of a refinable function vector and#934;(x) depend on the corresponding refinement newlinemask H(and#958; ). H(and#958; ) is a trigonometric matrix polynomial. Replacing eand#8722;iand#958; by z we get the newlinecorresponding matrix polynomial. Associated with a matrix polynomial there exist pairs newlineof matrices known as standard pair, left standard pair, decomposable pair, Jordan pair, newlineetc., which gives the spectral information about the corresponding matrix polynomial. newlineLiterature contains works which shows the construction of a refinement mask newlineand the corresponding multiwavelets using standard pairs and Jordan pairs. A refinable newlinefunction vector is usually construct