Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1501.07415

On $\ell^1$-regularization in light of Nashed's ill-posedness concept

Jens Flemming·Bernd Hofmann·Ivan Veselic
View original source ↗Read source document ↗Export metadata ↓

Abstract

Based on the powerful tool of variational inequalities, in recent papers convergence rates results on $\ell^1$-regularization for ill-posed inverse problems have been formulated in infinite dimensional spaces under the condition that the sparsity assumption slightly fails, but the solution is still in $\ell^1$. In the present paper we improve those convergence rates results and apply them to the Cesáro operator equation in $\ell^2$ and to specific denoising problems. Moreover, we formulate in this context relationships between Nashed's types of ill-posedness and mapping properties like compactness and strict singularity.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jens Flemming, Bernd Hofmann, Ivan Veselic. 2015-05-15. On $\ell^1$-regularization in light of Nashed's ill-posedness concept. https://doi.org/10.1515/cmam-2015-0008

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Complex Geometric Approach to the Discrete Gabor Transform and Localization Operators on the Flat Torus

In a recent paper, the discrete Gabor transform was connected to a Gabor transform with a time frequency domain given by the flat torus. We show that the corresponding Bargmann-Fock spaces can be expressed as theta functions (or equivalently line bundles on Abelian varieties). We give applications of this viewpoint to frame results for the discrete Gabor transform. In particular, we get necessary conditions which hold in higher dimensions and can expand the known results in the one dimensional case, the primary tool being the theorem of the square. We also give an application to asymptotics of restriction operators which arises via the asymptotic behavior of Bergman kernels and Toeplitz operators for high tensor powers of line bundles and find that time frequency restriction operators on the flat torus will exhibit "plunge" behaviors similar to those of time frequency restriction operators in other contexts.

math.FA↗
arXiv2026

On a minimal Andô dilation for a pair of strict contractions

The isometric dilation of a pair of commuting contractions due to Andô is not minimal. We modify Andô's dilation and construct a minimal isometric dilation on $\mathcal H \oplus_2 \ell_2(\mathcal H \oplus_2 \mathcal H)$ for a commuting pair of strict contractions on a Hilbert space $\mathcal H$. In the same spirit, we construct under certain conditions a minimal Andô dilation for a commuting pair of strict Banach space contractions. Further, we show that an Andô dilation is possible even for a more general pair of commuting contractions $(T_1,T_2)$ on a normed space $\mathbb X$ provided that the function $A_{T_i}: \mathbb X \rightarrow \mathbb R$ given by $A_{T_i}(x)=(\|x\|^2-\|T_ix\|^2)^{\frac{1}{2}}$ defines a norm on $\mathbb X$ for $i=1,2$.

math.FA↗
arXiv2026

Some properties of Fourier quasicrystals and measures on a strip

We extend certain results of the theory of Fourier quasicrystals on the real line to the case of a horizontal strip of finite width. For measures in a strip we use a natural generalization of the usual Fourier transform for measures on the line. We consider positive or translation bounded measures $μ$ on a strip whose Fourier transform is a pure point measure $\hatμ=\sum_{γ\inΓ}b_γδ_γ$ (as usual, $δ_γ$ is the unit mass at the point $γ$). We prove that the measure $ν=\sum_{γ\inΓ}|b_γ|^2δ_γ$ has the exponential growth. Moreover, if for some $η>0$ the points of $Γ$ in every interval of length $η$ are linearly independent over integers, then the measure $\hatμ$ also has the exponential growth.

math.FA↗