The cardinal orthogonal scaling function and sampling theorem in the wavelet subspaces

In this paper, we derive that there is a relation between the lowpass filter coefficient and wavelet’s samples in its integer points when a scaling function is a cardinal orthogonal scaling function. And we give some examples in which the lowpass filter coefficients are constructed from the wavelets. Then, we discuss the symmetry property of cardinal orthogonal scaling function, and give some useful characterizations. Therefore, we construct a family of cardinal orthogonal scaling functions with exponential decay and higher approximation order. Furthermore, some examples with exponential decay and higher approximation order are given, and they will testify our results. In the end, we deduce that there no exists a cardinal orthogonal scaling function with exponential decay, high approximation order and symmetry property in this family of cardinal orthogonal scaling functions constructed in our paper.