Construction For Wavelets Based On Algebra Method And Achieving Graph

There are many methods of construction wavelets and mutiwavelets, such as spectral factorization method. This paper construct wavelet bases using only the knowledge of algebra. So the construction wavelets and multiwavelets can be transformed to be linear algebraic problems.The paper discuss parametrizations of filters corresponding to with several vanishing moments using Grobner bases. After recalling some properties of biorthogonal wavelets, relations between the number of filters, symmetry, vanishing moments and discrete moments are discussed. The paper give up some vanishing moment conditions, which correspond to linear constraints on the filters, and introduce discrete moments of the filters as parameters. The resulting parametrized polynomial equations for the filters are solved by using Grobner bases. Finally, several different examples with a varying number of filters and vanishing moments are discussed in detail, explicit parametrizations of the filters are given including 7/9 symmetric biorthogonal wavelets. Then the paper discuss interpolatory multiscaling functions. Two-scale matrix symbol associated with interpolatory multiscaling functions is reduced to a special form. Also a new characterization on approximation order for the multiscaling functions is described in terms of elements of this special two-scale matrix symbol. An algorithm is provided for constructing compactly supported interpolating multiscaling functions with dilation factor 3 and higher approximation order. Finally, the associated several families examples with one-parameter or two-parameters are presented explicitly.This paper compute some examples using algorithm, then the paper obtain greater flexibility wavelets and multiwavelets, the algorithm is easy to be understood and generalize, the process of computing are easier. And the wavelets and multiwavelets are useful. Special parameter values that correspond to the scaling functions with the maximum Sobolev exponent are computed, then Sobolev exponent are given and the corresponding graphs of smooth scaling functions and wavelets are drew.