Wavelet Approximation in Weighted Sobolev Spaces of Mixed Order with Applications to the Electronic Schrödinger Equation

We study the approximation of functions in weighted Sobolev spaces of mixed order by anisotropic tensor products of biorthogonal, compactly supported wavelets. As a main result, we characterize these spaces in terms of wavelet coefficients, which also enables us to explicitly construct approximations. In particular, we derive approximation rates for functions in exponentially weighted Sobolev spaces discretized on optimized general sparse grids. Under certain regularity assumptions, the rate of convergence is independent of the number of dimensions. We apply these results to the electronic Schrödinger equation and obtain a convergence rate which is independent of the number of electrons; numerical results for the helium atom are presented.