Homogeneous Approximation Property for Multivariate Continuous Wavelet Transforms

The homogeneous approximation property (HAP) for the continuous wavelet transform is useful in practice because it means that the measure of the building area involved in a reconstruction of a function up to some error is essentially invariant under timescale shifts. For the univariate case, it was shown that the pointwise HAP holds if and only if the Fourier transforms of both wavelets and the function to be reconstructed are compactly supported on ??{0}. In this paper, we study the HAP for multivariate wavelet transforms. We show that similar results hold for this case. However, the above condition is only sufficient but not necessary if the dimension of the variable is greater than 1, which is different from the univariate case. We also get a convergence result on the inverse of wavelet transforms, which improves similar results by Daubechies and Holschneider and Tchamitchain.