Related papers: Hilbert spaces built on a similarity and on dynami…

We study the geometric structure of the reproducing kernel Hilbert space associated to the continuous wavelet transform generated by the irreducible representations of the Euclidean Motion $SE(2)$. A natural Hilbert norm for functions on…

In this paper we study the Hilbert space structure underlying the Koopman-von Neumann (KvN) operatorial formulation of classical mechanics. KvN limited themselves to study the Hilbert space of zero-forms that are the square integrable…

The possibility of defining sesquilinear forms starting from one or two sequences of elements of a Hilbert space is investigated. One can associate operators to these forms and in particular look for conditions to apply representation…

This note deals with a problem of the probabilistic Ramsey theory in functional analysis. Given a linear operator $T$ on a Hilbert space with an orthogonal basis, we define the isomorphic structure $\Sigma(T)$ as the family of all subsets…

We introduce a simple, general, and convergent scheme to compute generalized eigenfunctions of self-adjoint operators with continuous spectra on rigged Hilbert spaces. Our approach does not require prior knowledge about the eigenfunctions,…

We provide a detailed description of the model Hilbert space $L^2(\bbR; d\Sigma; \cK)$, were $\cK$ represents a complex, separable Hilbert space, and $\Sigma$ denotes a bounded operator-valued measure. In particular, we show that several…

Classifying points in high dimensional spaces is a fundamental geometric problem in machine learning. In this paper, we address classifying points in the $d$-dimensional Hilbert polygonal metric. The Hilbert metric is a generalization of…

Two ubiquitous aspects of large-scale data analysis are that the data often have heavy-tailed properties and that diffusion-based or spectral-based methods are often used to identify and extract structure of interest. Perhaps surprisingly,…

In this paper we initiate the study of a fundamental yet untapped random model of non-selfadjoint, bounded linear operators acting on a separable complex Hilbert space. We replace the weights $w_n=1$ in the classical unilateral shift $T$,…

We study a holographic theory of general spacetimes that does not rely on the existence of asymptotic regions. This theory is to be formulated in a holographic space. When a semiclassical description is applicable, the holographic space is…

Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a…

We present and study a novel class of one-dimensional Hilbert space eigenfunction transforms that diagonalize analytic difference operators encoding the (reduced) two-particle relativistic hyperbolic Calogero-Moser dynamics. The scattering…

Let $\mathscr T=(V, \mathcal E)$ be a leafless, locally finite rooted directed tree. We associate with $\mathscr T$ a one parameter family of Dirichlet spaces $\mathscr H_q~(q \geqslant 1)$, which turn out to be Hilbert spaces of…

We derive and analyse a new variant of the iteratively regularized Landweber iteration, for solving linear and nonlinear ill-posed inverse problems. The method takes into account training data, which are used to estimate the interior of a…

In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert…

This paper provides some first steps in developing empirical process theory for functions taking values in a vector space. Our main results provide bounds on the entropy of classes of smooth functions taking values in a Hilbert space, by…

A Hilbert space frame on $R^n$ is {\it scalable} if we can scale the vectors to make them a tight frame. There are known classifications of scalable frames. There are two basic questions here which have never been answered in any $R^n$:…

The theory of dynamical frames evolved from practical problems in dynamical sampling where the initial state of a vector needs to be recovered from the space-time samples of evolutions of the vector. This leads to the investigation of…

We explain why and how the Hilbert space comes about in quantum theory. The axiomatic structures of vector space, of scalar product, of orthogonality, and of the linear functional are derivable from the statistical description of quantum…

The generalized spectral theory is an effective approach to analyze a linear operator on a Hilbert space $\mathcal{H}$ with a continuous spectrum. The generalized spectrum is computed via analytic continuations of the resolvent operators…