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The degree of ill-posedness for some composition governed by the Cesaro operator

Yu Deng, Hans-Jürgen Fischer, Bernd Hofmann

TL;DR

We address the problem of determining the degree of ill-posedness for the composition $A=C∘J$ on $L^2(0,1)$, where $J$ is compact and $C$ is non-compact. The approach relates $A$ to the twofold integration operator $J^2$, analyzes the kernel of $A^*A$, and applies Hilbert-Schmidt based bounds with Legendre polynomials to sharpen the decay of the singular values. The main result is that $σ_n(A)\asymp n^{-2}$, establishing the exact ill-posedness degree $μ(A)=2$. This demonstrates that the Cesàro outer operator can increase the degree of ill-posedness for this class of compositions, and it provides insight into regularization needs for related inverse problems, complementing prior findings with other outer operators.

Abstract

In this article, we consider the singular value asymptotics of compositions of compact linear operators mapping in the real Hilbert space of quadratically integrable functions over the unit interval. Specifically, the composition is given by the compact simple integration operator followed by the non-compact Ces`aro operator possessing a non-closed range. We show that the degree of ill-posedness of that composition is two, which means that the Ces`aro operator increases the degree of illposedness by the amount of one compared to the simple integration operator.

The degree of ill-posedness for some composition governed by the Cesaro operator

TL;DR

We address the problem of determining the degree of ill-posedness for the composition on , where is compact and is non-compact. The approach relates to the twofold integration operator , analyzes the kernel of , and applies Hilbert-Schmidt based bounds with Legendre polynomials to sharpen the decay of the singular values. The main result is that , establishing the exact ill-posedness degree . This demonstrates that the Cesàro outer operator can increase the degree of ill-posedness for this class of compositions, and it provides insight into regularization needs for related inverse problems, complementing prior findings with other outer operators.

Abstract

In this article, we consider the singular value asymptotics of compositions of compact linear operators mapping in the real Hilbert space of quadratically integrable functions over the unit interval. Specifically, the composition is given by the compact simple integration operator followed by the non-compact Ces`aro operator possessing a non-closed range. We show that the degree of ill-posedness of that composition is two, which means that the Ces`aro operator increases the degree of illposedness by the amount of one compared to the simple integration operator.
Paper Structure (4 sections, 5 theorems, 58 equations, 2 figures)

This paper contains 4 sections, 5 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Cross connections to the twofold integration operator
  3. Hilbert-Schmidt property and kernel smoothness
  4. Improved upper bounds

Key Result

Lemma 1

The injective bounded linear operator $C: L^2(0,1) \to L^2(0,1)$ from eq:C is non-compact and not continuously invertible, i.e., the inverse operator $C^{-1}: \mathcal{R}(C) \subset L^2(0,1) \to L^2(0,1)$ is unbounded and hence the range $\mathcal{R}(C)$ is not a closed subset of $L^2(0,1)$.

Figures (2)

  • Figure 1: Double logarithmic representation of the computationally approximated eigenvalues $\lambda_n(A^*A)$ for small indices $n$.
  • Figure 2: Singular values of operators $A, J,$ and $J^2$

Theorems & Definitions (9)

  • Definition 1
  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Lemma 3
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2