Regularity of refinable functions with exponentially decaying masks

The regularity of refinable functions is an important issue in all multiresolution analysis and has a strong impact on applications of wavelets to image processing, geometric and numerical solutions of elliptic partial differential equations. The purpose of this paper is to characterize the regularity of refinable functions with exponentially decaying masks and a dilation matrix whose eigenvalues have the same modulus. The main results of this paper are really extensions of some results in Cohen et al. (1999) [5], Jia (1999) [17] and Lorentz and Oswald (2000) [28].