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The degree of ill-posedness of composite linear ill-posed problems with focus on the impact of the non-compact Hausdorff moment operator

Bernd Hofmann, Peter Mathé

TL;DR

This work analyzes composite linear ill-posed problems of the form $A=B\circ D$ with a compact inner factor $D$ and a non-compact outer factor $B$ having non-closed range, focusing on how $B$ influences the decay of singular values. By combining general $s$-number theory with conditional stability via index functions $\Psi$, the authors apply the framework to Hausdorff-moment compositions $B^{(H)}\circ \mathcal{E}^{(k)}$ and $B^{(H)}\circ J$, deriving logarithmic-type stability bounds and corresponding lower bounds for the singular values. They prove an improved upper bound $\sigma_i(B^{(H)}\circ J) \le C i^{-3/2}$, and thus $\sigma_i(B^{(H)}\circ J)/\sigma_i(J)=O(i^{-1/2})$, showing that a non-compact outer operator can increase the effective ill-posedness in this setting. A notable gap remains between these upper bounds and existing exponential-type lower bounds, attributed to kernel smoothness and Hilbert-matrix conditioning, leaving open questions about the exact decay rates and potential improvements via alternative bases. The results illuminate the degree of ill-posedness for composite inverse problems involving Hausdorff moments and guide regularization considerations in such settings.

Abstract

We consider compact composite linear operators in Hilbert space, where the composition is given by some compact operator followed by some non-compact one possessing a non-closed range. Focus is on the impact of the non-compact factor on the overall behaviour of the decay rates of the singular values of the composition. Specifically, the composition of the compact integration operator with the non-compact Hausdorff moment operator is considered. We show that the singular values of the composition decay faster than the ones of the integration operator, providing a first example of this kind. However, there is a gap between available lower bounds for the decay rate and the obtained result. Therefore we conclude with a discussion.

The degree of ill-posedness of composite linear ill-posed problems with focus on the impact of the non-compact Hausdorff moment operator

TL;DR

This work analyzes composite linear ill-posed problems of the form with a compact inner factor and a non-compact outer factor having non-closed range, focusing on how influences the decay of singular values. By combining general -number theory with conditional stability via index functions , the authors apply the framework to Hausdorff-moment compositions and , deriving logarithmic-type stability bounds and corresponding lower bounds for the singular values. They prove an improved upper bound , and thus , showing that a non-compact outer operator can increase the effective ill-posedness in this setting. A notable gap remains between these upper bounds and existing exponential-type lower bounds, attributed to kernel smoothness and Hilbert-matrix conditioning, leaving open questions about the exact decay rates and potential improvements via alternative bases. The results illuminate the degree of ill-posedness for composite inverse problems involving Hausdorff moments and guide regularization considerations in such settings.

Abstract

We consider compact composite linear operators in Hilbert space, where the composition is given by some compact operator followed by some non-compact one possessing a non-closed range. Focus is on the impact of the non-compact factor on the overall behaviour of the decay rates of the singular values of the composition. Specifically, the composition of the compact integration operator with the non-compact Hausdorff moment operator is considered. We show that the singular values of the composition decay faster than the ones of the integration operator, providing a first example of this kind. However, there is a gap between available lower bounds for the decay rate and the obtained result. Therefore we conclude with a discussion.

Paper Structure

This paper contains 7 sections, 13 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results for general operators
  3. Compositions with the Hausdorff moment operator
  4. Discussion of kernel smoothness
  5. Bounding the singular values of the composite operator $B^{(H)} \circ J$
  6. Discussion
  7. Proof of Proposition \ref{['pro:kernel']}

Key Result

Theorem 1

Let $D: X \to Z$ and $A: X \to Y$ be compact linear operators between the infinite dimensional Hilbert spaces $X,\,Y$ and $Z$ with non-closed ranges $\mathcal{R}(D)$ and $\mathcal{R}(A)$. Suppose that there exists an index function $\Psi:(0,\infty) \to (0,\infty)$ such that for $0<\delta \le \|A\|_{ holds. Then we have and also If the operators $D^{\ast}D\colon X \to X$ and $A^{\ast}A\colon X \t

Figures (1)

  • Figure 1: Plot of the kernel $k$ and its derivative $k_s$

Theorems & Definitions (28)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 18 more