Construction For3-band Biorthogonal Scaling Coefficients And Biorthogonal Multiwavelet

A symbolic computation method for moments with filter coefficientsof scaling functions is descried and parametrizing compactly supportedorthonormal wavelets is obtained. Following the idea, we are devoted to astudy moments and parameterization construction for3-band biorthogonalscaling coefficients with several vanishing moments. Firstly weinvestigate the relations between filter lengths and symmetry. Then, weprove the relationship between dual discrete moments of3-bandbiorthogonal scaling functions. This theorem reveals that the sum ofcontinuous moments of dual scaling functions φ and φ is completelydetermined by the lower discrete moments. And we show and prove thefact that the odd-indexed discrete moments are determined by the smallereven-indexed discrete moments. Finally, a family3-band biorthogonalscaling coefficients with discrete moments as parameters are explicitlyexpressed based on computer algebra. And we use symbolic computationmethods which were introduced to solve polynomial equations for the filtercoefficients.In the second part of our paper, We focuses on the construction ofbiorthogonal interpolating multiscaling functions of multiplicity four withapproximation order N. Firstly using interpolating property, the parametricexpression with filters of the biorthogonal multiscaling function is derived.Secondly, this paper also show that there no exists any interpolatingbiorthogonal multiwavelet of multiplicity4with symmetry. And we described the relationship between dual filters of multiscaling functions.Then, the low-pass filters can be found by solving the equations generated.