Global solutions of approximation problems in Hilbert spaces

We study three well-known minimization problems in Hilbert spaces: the weighted least squares problem and the related problems of abstract splines and smoothing. In each case, we analyse the solvability of the problem for every point of the Hilbert space in the corresponding data set, the existence of an operator that maps each data point to its solution in a linear and continuous way, and the solvability of the associated operator problem in a fixed p-Schatten norm.