Global solutions of approximation problems in Hilbert spaces

Abstract

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 analyze 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.Keywords: Abstract spline problems, Schatten p lasses, optimal inverses 2000 MSC: 47A05, 46C05, 41A65

Total least squares problems on infinite dimensional spaces

“…We present necessary conditions for a pair (A 0 , x 0 ), to be a solution of the RWTLS problem. We observe that if the RWTLS problem has a solution (A 0 , x 0 ), then x 0 is a solution of the classical smoothing problem [3,9,10,13,14] and the results obtained in [11] can be applied for giving necessary conditions for the existence of solution of problem (1.2). In section 3.3, we study the case where the regularization is given by a multiple of the identity operator, ρI.…”