Band-limited Wavelets and Framelets in Low Dimensions

In this paper, we study the problem of constructing non-separable band-limited wavelet tight frames, Riesz wavelets and orthonormal wavelets in $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$ . We first construct a class of non-separable band-limited refinable functions in low-dimensional Euclidean spaces by using univariate Meyer’s refinable functions along multiple directions defined by classical box-spline direction matrices. These non-separable band-limited definable functions are then used to construct non-separable band-limited wavelet tight frames via the unitary and oblique extension principles. However, these refinable functions cannot be used for constructing Riesz wavelets and orthonormal wavelets in low dimensions as they are not stable. Another construction scheme is then developed to construct stable refinable functions in low dimensions by using a special class of direction matrices. The resulting stable refinable functions allow us to construct a class of MRA-based non-separable band-limited Riesz wavelets and particularly band-limited orthonormal wavelets in low dimensions with small frequency support.