Multiresolution Analysis for Compactly Supported Interpolating Tensor Product Wavelets (long version)

Abstract

We construct multidimensional interpolating tensor product MRA's of the function spaces $C_0(\mathbb{R}^n,K)$, $K = \mathbb{R}$ or $K = \mathbb{C}$, consisting of real or complex valued functions on $\mathbb{R}^n$ vanishing at infinity and the function spaces $C_\textrm{u}(\mathbb{R}^n,K)$ consisting of bounded and uniformly continuous functions on $\mathbb{R}^n$. We also construct an interpolating dual MRA for both of these spaces. The theory of the tensor products of Banach spaces is used. We generalize the Besov space norm equivalence result from Donoho (1992, Interpolating Wavelet Transforms) to our $n$-dimensional construction.