Best N Term Approximation Spaces for Tensor Product Wavelet Bases

摘要

We consider best N term approximation using anisotropic tensor product wavelet bases ("sparse grids"). We introduce a tensor product structure ⊗q on certain quasi-Banach spaces. We prove that the approximation spaces Aα q(L2) and Aα q(H1) equal tensor products of Besov spaces Bα q(Lq), e.g., Aα q(L2([0,1]d)) = Bα q(Lq([0,1])) ⊗q · ⊗q Bα q · ·(Lq([0,1])). Solutions to elliptic partial differential equations on polygonal/polyhedral domains belong to these new scales of Besov spaces.