Wavelet analysis provides suitable bases for the class of L 2 functions. The function to be represented is approximated at different resolutions. The desirable properties of a basis are orthogonality, compact supportedness and symmetricity. In the scalar case, the only wavelet with these properties is Haar wavelet. Theory of multiwavelets assumes significance since it offers symmetric, compactly supported, orthogonal bases for L 2 .R/. The properties of a multiwavelet are determined by the corresponding Multiscaling Function. A multiscaling function is characterized by its symbol function which is a matrix polynomial in complex exponential. The inverse representation theorem of matrix polynomials provides a method to construct a matrix polynomial from its Jordan pair. Our objective is to find the properties that characterize a Jordan pair of a symbol function of a multiscaling function with desirable properties.
Wavelet analysis deals with finding a suitable basis for the class of L 2 functions. Symmetric basis functions are very useful in various applications. In the case of all wavelets other than the famous Haar wavelet, the simultaneous inclusion of compact supportedness, orthogonality and symmetricity is not possible. Theory of multiwavelets assumes significance since it offers orthogonal, compact frames without losing symmetry. We can also construct symmetric, compactly supported and pseudo-biorthogonal bases which are also possible only in the case of multiwavelets. The properties of a multiwavelet directly depends on the corresponding multiscaling function. A multiscaling function is characterized by a unique symbol function, which is a matrix polynomial in complex exponential. A matrix polynomial can be constructed from its spectral data. Our aim is to find the necessary as well as sufficient conditions a spectral data must satisfy so that the corresponding matrix polynomial is the symbol function of a compactly supported, symmetric multiscaling function UðxÞ. We will construct such a multiscaling function UðxÞ and its dualŨðxÞ such that the functions UðxÞ andŨðxÞ form a pair of pseudo-biorthogonal multiscaling functions.
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