Orthogonal finite element and multi-resolution theory

Finite element can approach any function to any accuracy. Spaces of finite element function of rough division are included in the space of element function of fine division, therefore one can obtain a multi-resolution theory by construction of relationship among spaces of different division finite element function. The multi-resolution theory bases on the usually finite element theory will result in great calculation. Representation theory of group is used to build the orthogonal bases of finite element spaces and by this method a multi-resolution theory based on finite element method is developed. The theory can be applied to signal processing.