Related papers: A Generalized Sampling Theorem for Stable Reconstr…
Generalized sampling is a recently developed linear framework for sampling and reconstruction in separable Hilbert spaces. It allows one to recover any element in any finite-dimensional subspace given finitely many of its samples with…
We introduce a method to reconstruct an element of a Hilbert space in terms of an arbitrary finite collection of linearly independent reconstruction vectors, given a finite number of its samples with respect to any Riesz basis. As we…
Generalized sampling consists in the recovery of a function $f$, from the samples of the responses of a collection of linear shift-invariant systems to the input $f$. The reconstructed function is typically a member of a finitely generated…
We prove a sampling theorem for infinite-dimensional Paley-Wiener spaces on graphs which allows for stable frame reconstruction. We prove that all sampling sets for a fixed Paley-Wiener space are complements of lambda-sets (i.e. sets where…
Reconstructing an infinite-dimensional signal from a finite set of measurements is a fundamental problem in approximation theory and signal processing. While the generalized sampling (GS) framework provides a robust methodology for…
This paper introduces a novel framework and corresponding methods for sampling and reconstruction of sparse signals in shift-invariant (SI) spaces. We reinterpret the random demodulator, a system that acquires sparse bandlimited signals, as…
An axiomatic approach to signal reconstruction is formulated, involving a sample consistent set and a guiding set, describing desired reconstructions. New frame-less reconstruction methods are proposed, based on a novel concept of a…
A useful sampling-reconstruction model should be stable with respect to different kind of small perturbations, regardless whether they result from jitter, measurement errors, or simply from a small change in the model assumptions. In this…
The reconstruction theorem, a cornerstone of Martin Hairer's theory of regularity structures, appears in this article as the unique extension of the explicitly given reconstruction operator on the set of smooth models due its inherent…
We establish a regular sampling theory in the range of the analysis operator of a continuous frame having a unitary structure. The unitary structure is related with a unitary representation of a locally compact abelian group on a separable…
The theory of sampling and the reconstruction of data has a wide range of applications and a rich collection of techniques. For many methods a core problem is the estimation of the number of samples needed in order to secure a stable and…
This paper explores woven frames in separable Hilbert spaces with an initial focus on the finite-dimensional case. We begin by simplifying the problem to bases, for which we obtain a unique characterization. We establish a condition that is…
Signal sampling and reconstruction is a fundamental engineering task at the heart of signal processing. The celebrated Shannon-Nyquist theorem guarantees perfect signal reconstruction from uniform samples, obtained at a rate twice the…
Given an orthonormal basis $ {\mathcal V}= \{v_j\} _{j\in N}$ in a separable Hilbert space $H$ and a set of unit vectors $ {\mathcal B}=\{w_j\}_{j\in N}$, we consider the sets $ {\mathcal B}_N$ obtained by replacing the vectors $v_1, ...,\,…
Weighted average sampling is more practical and numerically more stable than sampling at single points as in the classical Shannon sampling framework. Using the frame theory, one can completely reconstruct a bandlimited function from its…
Sampling theory has traditionally drawn tools from functional and complex analysis. Past successes, such as the Shannon-Nyquist theorem and recent advances in frame theory, have relied heavily on the application of geometry and analysis.…
Two years ago, Conlon and Gowers, and Schacht proved general theorems that allow one to transfer a large class of extremal combinatorial results from the deterministic to the probabilistic setting. Even though the two papers solve the same…
Frame theory provides a robust method for recovering vectors in a Hilbert space from inner product data, though the associated decomposition formula can be computationally demanding. We relax the frame condition by studying sequences that…
We develope a local theory for frames on finite dimensional Hilbert spaces. In particular, a bounded frame on a finite dimensional Hilbert space contains a subset which is a good Riesz basis for a percentage (arbitrarily close to one) of…
In the field of signal processing, the sampling theorem plays a fundamental role for signal reconstruction as it bridges the gap between analog and digital signals. Following the celebrated Nyquist-Shannon sampling theorem, generalizing the…