Related papers: The Frame Potential, on Average
Every frame in Hilbert space contains a subsequence equivalent to an orthogonal basis. If a frame is n-dimensional then this subsequence has length (1 - \epsilon) n. On the other hand, there is a frame which does not contain bases with…
Sequences of unit vectors for which the Kaczmarz algorithm always converges in Hilbert space can be characterized in frame theory by tight frames with constant 1. We generalize this result to the context of frames and bases. In particular,…
The objects of the present study are one-parameter semigroups generated by Schr\"odinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector…
Finite frames can be viewed as mass points distributed in $N$-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of…
We introduce the notion of continuous frame in n-Hilbert space which is a generalization of discrete frame in n-Hilbert space. The tensor product of Hilbert spaces is a very important topic in mathematics. Here we also introduce the concept…
The ability of the Rigged Hilbert Space formalism to deal with continuous spectrum is demonstrated within the example of the square barrier potential. The non-square integrable solutions of the time-independent Schrodinger equation are used…
Given a frame in a finite dimensional Hilbert space we construct additive perturbations which decrease the condition number of the frame. By iterating this perturbation, we introduce an algorithm that produces a tight frame in a finite…
This paper provides sufficient density conditions for the existence of smooth vectors generating a frame or Riesz sequence in the lattice orbit of a square-integrable projective representation of a nilpotent Lie group. The conditions…
We consider frames in a finite-dimensional Hilbert space Hn where frames are exactly the spanning sets of the vector space. We present a method to determine the maximum robustness of a frame. We present results on tight subframes and…
In order to extend the spectral action principle to non-compact spaces, we propose a framework for spectral triples where the algebra may be non-unital but the resolvent of the Dirac operator remains compact. We show that an example is…
The frame set of a function $g\in L^2(\mathbb{R})$ is the set of all parameters $(a, b)\in \mathbb{R}^2_+$ for which the collection of time-frequency shifts of $g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for…
The concept of operator frame can be considered as a generalization of frame. Firstly, we introduce the notion of operator frame for the set of all adjointable operators $Hom_{\mathcal{A}}^{\ast}(\mathcal{X})$ on a Hilbert…
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized…
Let $\mathcal F_0=\{f_i\}_{i\in\mathbb{I}_{n_0}}$ be a finite sequence of vectors in $\mathbb C^d$ and let $\mathbf{a}=(a_i)_{i\in\mathbb{I}_k}$ be a finite sequence of positive numbers. We consider the completions of $\cal F_0$ of the form…
We introduce a sl_2-invariant family of nonlinear vector fields with a non-semisimple triple zero singularity. In this paper we are concerned with characterization and normal form classification of these vector fields. We show that the…
Given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we describe the spectral and geometrical structure of optimal completions of $\mathcal F_0$ obtained by adding a finite sequence of vectors with prescribed norms, where optimality…
A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a non-degenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to…
Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries…
Let M be a compact Riemannian manifold without boundary and let E be a Riemannian vector bundle over M. If $\sigma$ denotes the sphere subbundle of E, we look for embeddings of $\sigma$ into E admitting a prescribed mean curvature.
We apply ourselves to the noncommutative geometry of frame bundles by showing that each C$^*$-algebraic noncommutative principal $\mathrm{SO}(n)$-bundle is, up to isomorphism, uniquely determined by its associated noncommutative vector…