Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the type of ill-posedness of generalized Hilbert matrices and related operators

Stefan Kindermann

Abstract

We consider infinite-dimensional generalized Hilbert matrices of the form $H_{i,j} = \frac{d_i d_j}{x_i + x_j}$, where $d_i$ are nonnegative weights and $x_i$ are pairwise disjoint positive numbers. We state sufficient and, for monotonically rearrangeable $x_i$, also necessary conditions for $d_i$, $x_i$ such that the induced operator from $\ell^2 \to \ell^2$ and related operators are well-defined, bounded, or compact. Furthermore, we give conditions, when this operator is injective and ill-posed.

On the type of ill-posedness of generalized Hilbert matrices and related operators

Abstract

We consider infinite-dimensional generalized Hilbert matrices of the form , where are nonnegative weights and are pairwise disjoint positive numbers. We state sufficient and, for monotonically rearrangeable , also necessary conditions for , such that the induced operator from and related operators are well-defined, bounded, or compact. Furthermore, we give conditions, when this operator is injective and ill-posed.
Paper Structure (11 sections, 9 theorems, 82 equations)

This paper contains 11 sections, 9 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Forward properties
  3. Related operators
  4. Main result
  5. Proof of well-posedness
  6. Proof of boundedness
  7. Proof of compactness
  8. Inverse properties
  9. Injectivity
  10. Examples and Consequences
  11. Singular values of $\mathcal{H}$

Key Result

Theorem 1

For $d_i,x_i$ as stated above, define for $L\in \mathbb{N}$ Then the operator $\mathcal{H}:\ell^2 \to \ell^2$ is In case that $x_i$ is monotonically rearrangeable, we have that $\mathcal{H}$ is

Theorems & Definitions (17)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • ...and 7 more