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Global solutions of approximation problems in Krein spaces

Maximiliano Contino

Abstract

Three approximation problems in Krein spaces are studied, namely the indefinite weighted least squares problem and the related problems of indefinite abstract splines and smoothing. In every case, we analyze if the problem has a solution for every point of the Krein space, the existence of a linear and continuous operator that maps each data point to its solution and when the associated operator problem considering the J-trace has a solution.

Global solutions of approximation problems in Krein spaces

Abstract

Three approximation problems in Krein spaces are studied, namely the indefinite weighted least squares problem and the related problems of indefinite abstract splines and smoothing. In every case, we analyze if the problem has a solution for every point of the Krein space, the existence of a linear and continuous operator that maps each data point to its solution and when the associated operator problem considering the J-trace has a solution.
Paper Structure (4 sections, 13 theorems, 39 equations)

This paper contains 4 sections, 13 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Global solutions of weighted least squares problems
  4. Global solutions of spline and smoothing problems

Key Result

Proposition 3.1

Let $A\in L(\mathcal{K},\mathcal{H})$, $W\in L(\mathcal{H})^{s}$ and $x \in \mathcal{H}.$ Then $u \in \mathcal{K}$ is a $W$-ILSS of $Az =x$ if and only if $\mathop{\mathrm{ran}}\nolimits A$ is $W$-nonnegative and $A^{\#}W(Au-x)=0.$

Theorems & Definitions (23)

  • Definition
  • Definition
  • Definition
  • Proposition 3.1
  • Definition
  • Theorem 3.2
  • Corollary 3.3
  • Remark 1
  • Definition
  • Proposition 3.4
  • ...and 13 more