# Global solutions of approximation problems in Hilbert spaces

## TL;DR

This paper investigates fundamental minimization problems in Hilbert spaces, analyzing their solvability, the existence of solution operators, and properties related to Schatten norms, providing a comprehensive theoretical framework.

## Contribution

It offers a detailed analysis of three classical minimization problems in Hilbert spaces, including conditions for solvability and the existence of continuous solution operators.

## Key findings

- Characterization of solvability conditions for each problem
- Existence criteria for linear, continuous solution operators
- Analysis of operator problems in p-Schatten norms

## 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.