Tensor-product wavelet basis

From a one-dimensional multiresolution analysis, the simultaneous-scale two-dimensional detail space splits as $(W_j\otimes V_j)\oplus (V_j\otimes W_j)\oplus (W_j\otimes W_j)$. Its generating wavelets are $\psi \otimes \phi$, $\phi \otimes \psi$, and $\psi \otimes \psi$. Their normalized dilates are $2^j\Psi (2^j\cdot -k)$ for $k\in {\mathbb{Z}}^2$. Include a coarse scaling family when working on a bounded square.

A finite-length smooth curve meets $O(2^j)$ supports of localized two-dimensional wavelets at scale $j$. For a bounded function, their coefficients are $O(2^{-j})$, so their total squared energy at that scale is $O(2^{-j})$. Keeping these coefficients through level $J$ uses $O(2^J)$ terms and leaves squared error $O(2^{-J})$. Polynomial cancellation, or adequate approximation of the remaining smooth regions, therefore gives best N-term approximation squared error $O(N^{-1})$.

A tensor-product wavelet is a product of one-dimensional wavelets and scaling functions, with at least one wavelet factor. In two dimensions, the three simultaneous-scale types are $\psi \otimes \phi$, $\phi \otimes \psi$ and $\psi \otimes \psi$. Their translates and normalized dilates form a tensor-product wavelet basis.