Related papers: A solution to the polarization problem

The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…

In this paper, we prove optimal convergence rates results for regularisation methods for solving linear ill-posed operator equations in Hilbert spaces. The result generalises existing convergence rates results on optimality to general…

A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that if, in $\mathbb{R}^n$, the unit ball of some norm is equal to the unit ball of the dual norm, then the norm…

In this paper, we consider linear ill-posed problems in Hilbert spaces and their regularization via frame decompositions, which are generalizations of the singular-value decomposition. In particular, we prove convergence for a general class…

We consider positivity conditions both for real-valued functions of several complex variables and for Hermitian forms. We prove a stabilization theorem relating these two notions, and give some applications to proper mappings between balls…

An exact, number-conserving solution to the generalized, orbit-dependent pairing problem is derived by introducing an infinite-dimensional algebra. A method for obtaining eigenvalues and eigenvectors of the corresponding Hamiltonian is also…

We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. The construction starts with a positive selfadjoint operator $H$, that is called the Hamiltonian of the…

A new technique for proving fixed point theorems for families of holomorphic transformations of operator balls is developed. One of these theorems is used to show that a bounded representation in a real or complex Hilbert space is…

We present a Hilbert space perspective to homogenization of standard linear evolutionary boundary value problems in mathematical physics and provide a unified treatment for (non-)periodic homogenization problems in thermodynamics,…

In this paper, we bring a complete solution to the Ovals problem, as formulated in [3] and [24].

We present a reduction of the Hilbert-Smith conjecture in the case of the finite dimensional orbit space to some algebraic topology problems.

We propose a Hilbert space solution theory for a nonhomogeneous heat equation with delay in the highest order derivatives with nonhomogeneous Dirichlet boundary conditions in a bounded domain. Under rather weak regularity assumptions on the…

We characterize when a coherent state or continuous frame for a Hilbert space may be sampled to obtain a frame, which solves the discretization problem for continuous frames. In particular, we prove that every bounded continuous frame for a…

The hyperinvariant subspace problem is solved in the setting of Hilbert and right Hamilton space, motivated by my earlier works in the invariant subspace problem.

We prove that a large class of parabolic final value problems is well posed.This results via explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability of solutions. This data space is the graph normed…

This paper presents finite-time and fixed-time stabilization results for inhomogeneous abstract evolution problems, extending existing theories. We prove well-posedness for strong and weak solutions, and estimate upper bounds for settling…

In this paper, existence of pairs of solutions is obtained for compact potential operators on Hilbert spaces. An application to a second-order boundary value problem is also given as an illustration of our results.

We give a geometric characterization of extremal sets in ell_p spaces that generalizes our previous result for such sets in Hilbert spaces.

In a finite dimensional Hilbert space, each normalized vector (state) can be chosen as a member of an orthonormal basis of the space. We give a proof of this statement in a manner that seems to be more comprehensible for physics students…

Several convergence results in Hilbert scales under different source conditions are proved and orders of convergence and optimal orders of convergence are derived. Also, relations between those source conditions are proved. The concept of a…