Approximation properties of multivariate wavelets

Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in W 1 k − 1 ( R s ) W₁ᵏ⁻¹(Rˢ) provides approximation order k k .