Construction Of Two Kinds Of Scalar Function

In nearly three decades, wavelet analysis is a new and rapidly developed branch in the fields of mathematics and applied mathematics. The remarkable accomplish- ment in theory and application have been achieved. In the analysis of wavelet, the construction of wavelet plays an important role in the aspects such as research on the properties of wavelet and application. The method that the wavelet function or multi-wavelet function is conducted by means of multiresolution analysis is divided into two steps. The first step, the most essential step, is how to establish the scale function. The second step is how to construct the corresponding wavelet function by the scale function obtained in the first step. In this paper, a scale function with the limited one-dimensional frequency spectrum and a two-dimensional multi-scale functions are established, which provides a way to construct one-dimensional or two-dimensional scale function. The concrete work is introduced as following:Firstly, based on conducting a equivalence condition of limited and frequency spectral scale function, a method for constructing a scale function with the limited one-dimensional frequency spectrum is given. Finally, two cocrete examples are given in order to make the method be realized.Secondly, a bivariate orthogonal multi-scale function is constructed by n filters and a bivariate single orthogonal function. Based on the character of filters, several Equivalent conditions are obtained. Moreover, for the further study of finite pulse filters, it is shown that this construction is effective by constructing a specific example. The two-dimensional orthogonal multi-scale function constructed is practical and feasible in theory. As long as the selection of the filter satisfies certain conditions, the two-dimensional orthogonal multi-scale function obtained by this method can be constructed. Moreover, this method also has the flexibility.