"Using transformations to generalize approximation operators for periodic functions"
Lippert, LauraThere are several approximation operators for high-dimensional periodic functions available. We propose methods to transform functions on R^d or the cube [0, 1]^d to functions on T^d, apply an approximation based on hyperbolic wavelet regression and transform the resulting function back. We study the problem of scattered-data approximation, where we have given sample points and the corresponding function evaluations. We transform the sample points to points on T^d, evaluate the basis functions at these points, create a matrix and solve the matrix equation with an LSQR-algorithm to get an approximation. In our case this matrix is sparse, since we deal with compactly supported wavelets. If we choose the number of parameters such that we have logarithmic oversampling, we can give a lower bound for the norm of the Moore-Penrose inverse, so that the LSQR-algorithm gives useful results. For most approximation operators the error can be estimated well if we are concerned with uniformly i.i.d. samples. But with the transformation idea we can even deal with non-uniformly distributed data.