Functions with Prescribed Best Linear Approximations

P. L. Combettes; N. N. Reyes

Home/Archive/arXiv

arXiv · 0905.3520

Functions with Prescribed Best Linear Approximations

P. L. Combettes·N. N. Reyes

View original source ↗Read source document ↗Export metadata ↓

Abstract

A common problem in applied mathematics is to find a function in a Hilbert space with prescribed best approximations from a finite number of closed vector subspaces. In the present paper we study the question of the existence of solutions to such problems. A finite family of subspaces is said to satisfy the \emph{Inverse Best Approximation Property (IBAP)} if there exists a point that admits any selection of points from these subspaces as best approximations. We provide various characterizations of the IBAP in terms of the geometry of the subspaces. Connections between the IBAP and the linear convergence rate of the periodic projection algorithm for solving the underlying affine feasibility problem are also established. The results are applied to problems in harmonic analysis, integral equations, signal theory, and wavelet frames.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

P. L. Combettes, N. N. Reyes. 2009-05-21. Functions with Prescribed Best Linear Approximations. https://arxiv.org/abs/0905.3520

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

KEEP EXPLORING

Related papers

arXiv2026

Metric Poincaré type inequalities and lower bounds on the infimum of the spectrum for graphs

We study metric Poincaré type inequalities on general graphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants geometrically as diameters and inradii. Moreover, we can relate them to spectral theory of Laplacians once a probability measure on the graph is chosen. More specifically, we obtain a variational characterization of these constants as infimum over spectral gaps of all Laplacians on the graphs associated to probability measures

math.FA↗

arXiv2026

Natural methods of unsupervised topological alignment

In this paper, we consider methods for the diagonal multi-omics integration of heterogeneous datasets. Several approaches to the nature of biological heterogeneity are analyzed and developed to comprehend more clearly the generated differences. Specifically, the extremal trace problems for the coupled Laplacian on sets homeomorphic to the Stiefel manifold embedded in the complex Euclidean space are investigated. The gradient ascent method for the maximization problem is elaborated in the classical terms of functional analysis, which is of significant interest in itself. On this basis, we introduce a novel characteristic of dataset heterogeneity by employing the norm of the difference between the maximum and minimum points.

math.FA↗

arXiv2026

On Toeplitz operators on compact Abelian groups and discrete Wiener--Hopf operators

This paper introduces the concept of a rotation number for a continuous, non-degenerate two-dimensional vector field (a zero-free complex-valued function) on a compact connected Abelian group. This concept generalizes the notion of a finite rotation number for such groups, previously introduced by the author. Using this concept, a Gohberg-Krein index formula is derived for semi-Fredholm Toeplitz operators with continuous symbols defined on such groups. Criteria for these operators to be semi-Fredholm are established, and their essential spectra are described. As a by-product for the continuous symbol case, conditions for Fredholmness and semi-Fredholmness are established, and the Fredholm index of Wiener-Hopf operators over a linearly ordered discrete Abelian group is calculated in terms of their symbols. Spectral properties-including the spectra and essential spectra-of the Wiener-Hopf operators under consideration are also described.

math.FA↗