Related papers: Approximately Dual and Pseudo-Dual Probabilistic F…

This paper studies probabilistic dual frames and the associated dual frame potentials from the perspective of optimal mass transport. The main contribution of this work shows that given a probabilistic frame, its associated dual frame…

Functions or distributions used to sample and to reconstruct signals often occur in different domains, like the Dirac delta and a band-limited bump function in classical sampling. Oblique dual frames generalize this phenomenon. In this…

Probabilistic frames are a generalization of finite frames into the Wasserstein space of probability measures with finite second moment. We introduce new probabilistic definitions of duality, analysis, and synthesis and investigate their…

We study sparsity and spectral properties of dual frames of a given finite frame. We show that any finite frame has a dual with no more than $n^2$ non-vanishing entries, where $n$ denotes the ambient dimension, and that for most frames no…

We introduce probabilistic frames to study finite frames whose elements are chosen at random. While finite tight frames generalize orthonormal bases by allowing redundancy, independent, uniformly distributed points on the sphere…

In the paper Optimal Dual Frames for Probabilistic Erasures, the authors have given conditions under which the canonical dual is claimed to be the unique probability optimal dual for 1-erasure reconstruction. In this paper, we demonstrate…

We discuss the concepts of pseudo-dual frames and approximately dual frames, and illuminate their relationship to classical frames. Approximately dual frames are easier to construct than the classical dual frames, and might be tailored to…

Several finite dimensional quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These representations are often defined only on restricted…

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…

Fusion frames are valuable generalizations of discrete frames. Most concepts of fusion frames are shared by discrete frames. However, the dual setting is so complicated. In particular, unlike discrete frames, two fusion frames are not dual…

A probabilistic frame is a Borel probability measure with finite second moment whose support spans $\mathbb{R}^d$. A Parseval probabilistic frame is one for which the associated matrix of the second moments is the identity matrix in…

In this chapter we survey two topics that have recently been investigated in frame theory. First, we give an overview of the class of scalable frames. These are (finite) frames with the property that each frame vector can be rescaled in…

We consider the problem of rescaling the lengths of a finite frame thereby transforming it into a tight one. Such frames are called scalable and have received a lot of attention in recent years. In this note we investigate the question in…

Building on earlier work, we further develop a formalism based on the mathematical theory of frames that defines a set of possible phase-space or quasi-probability representations of finite-dimensional quantum systems. We prove that an…

In a previous paper [Adcock & Huybrechs, 2019] we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but…

A self-dual algebras is one isomorphic as a module to the opposite of its dual; a quasi self-dual algebra is one whose cohomology with coefficients in itself is isomorphic to that with coefficients in the opposite of its dual. For these…

In this work we introduce a statistical framework in order to analyze the spatial redundancy in natural images. This notion of spatial redundancy must be defined locally and thus we give some examples of functions (auto-similarity and…

Redundancy is the qualitative property which makes Hilbert space frames so useful in practice. However, developing a meaningful quantitative notion of redundancy for infinite frames has proven elusive. Though quantitative candidates for…

In representations using frames, oblique duality appears in situations where the analysis and the synthesis has to be done in different subspaces. In some cases, we cannot obtain an explicit expression for the oblique duals and in others…

We show that any two frames in a separable Hilbert space that are dual to each other have the same excess. Some new relations for the analysis resp. synthesis operators of dual frames are also derived. We then prove that pseudo-dual frames…