Compactly supported multivariate multiwavelets: Theory and constructions

A new sufficient condition for orthonormality of refinable functions is developed. More details are explored when the multiplicity {dollar}r=2{dollar} in the univariate and bivariate settings. This sufficient condition is applied to check orthonormality of refinable functions.; A class of univariate compactly supported orthonormal multi-scaling functions are constructed. Then a decay estimate for their Fourier transforms is given. We present a method to construct a multivariate refinable function from a univariate refinable function which is not of tensor product type. A class of multivariate non-tensor product compactly supported orthonormal multi-scaling functions with arbitrarily high regularity are constructed. We also describe how to do unitary matrix extension for our multi-scaling functions to find the multiwavelets.; Two classes of biorthogonal multi-scaling functions, one from B-splines and the other from the box splines, are constructed. Their stability, biorthogonality and regularity are also studied. We give specific non-singular matrix extensions to find the multiwavelets.