Related papers: Paley-Wiener Theorem for Probabilistic Frames
Notion of frames and Bessel sequences for metric spaces have been introduced. This notion is related with the notion of Lipschitz free Banach spaces. \ It is proved that every separable metric space admits a metric $\mathcal{M}_d$-frame.…
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…
Parseval and equal-norm frames play a fundamental role in frame theory and signal processing. In this work, we prove non-asymptotic concentration bounds showing that random equal-norm frames are nearly Parseval with high probability, and…
We extend the Paley--Wiener theorem for riemannian symmetric spaces to an important class of infinite dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit…
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…
We develop the theory of frames and Parseval frames for finite-dimensional vector spaces over the binary numbers. This includes characterizations which are similar to frames and Parseval frames for real or complex Hilbert spaces, and the…
In this paper we analyze the approximation of stable linear time-invariant systems, like the Hilbert transform, by sampling series for bandlimited functions in the Paley-Wiener space $\mathcal{PW}_{\pi}^{1}$. It is known that there exist…
Let S(X) be the Schwartz space of compactly supported smooth functions on the p-adic points of a spherical variety X, and let C(X) be the space of Harish-Chandra Schwartz functions. Under assumptions on the spherical variety, which are…
An $n \times n$ matrix with $\pm 1$ entries which acts on $\mathbb{R}^n$ as a scaled isometry is called Hadamard. Such matrices exist in some, but not all dimensions. Combining number-theoretic and probabilistic tools we construct matrices…
We consider existence and uniqueness of symmetric approximation of frames by normalized tight frames and of symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces H . More precisely, we determine whether a given frame…
The Bayesian perspective on inverse problems has attracted much mathematical attention in recent years. Particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an…
Existing concentration bounds for bounded vector-valued random variables include extensions of the scalar Hoeffding and Bernstein inequalities. While the latter is typically tighter, it requires knowing a bound on the variance of the random…
This paper is concerned with the implications of sufficient conditions ensuring that a perturbation of a frame is again a frame. We emphasize how stability of frames is fundamental for numerical applications and we discuss in particular the…
In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted…
Probabilistic argumentation is an alternative to causal modeling with Bayesian networks. Probabilistic argumentation structures (PAS) are defined on families of compatible frames (f.c.f). This is a generalization of the usual multivariate…
A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function…
A notion of Paley-Wiener spaces is introduced on combinatorial graphs. It is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets. Such…
We give necessary and sufficient conditions for a subfamily of regularly spaced translates of a function to form a frame (resp. a Riesz basis) for its span. One consequence is that ifthetranslates are taken only from a subset of the natural…
It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel…
Paley-Wiener type theorems describe the image of a given space of functions, often compactly supported functions, under an integral transform, usually a Fourier transform on a group or homogeneous space. Several authors have studied…