Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces

Hunt, Brian R.; Sauer, Tim; Yorke, James A.

doi:10.1090/s0273-0979-1992-00328-2

Cited by 364 publications

(382 citation statements)

References 30 publications

Supporting

Mentioning

372

Contrasting

Unclassified

Order By: Relevance

“…Hunt, T. Sauer and J.A. Yorke [5]- [6] found this notation again, but in a topological abelian group with a complete metric (not necessary separable).…”

Section: Introductionmentioning

confidence: 88%

Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

Jabłońska

2015

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

In the paper we would like to pay attention to some analogies between Haar meager sets and Haar null sets. Among others, we will show that 0 ∈ int (A− A) for each Borel set A, which is not Haar meager in an abelian Polish group. Moreover, we will give an example of a Borel non-Haar meager set A ⊂ c 0 such that int (A + A) = ∅. Finally, we will define D-measurability as a topological analog of Christensen measurability, and apply our generalization of Piccard's theorem to prove that each D-measurable homomorphism is continuous. Our results refer to the papers [1], [2] and [4].

show abstract

“…Hunt, T. Sauer and J.A. Yorke [5]- [6] found this notation again, but in a topological abelian group with a complete metric (not necessary separable).…”

Section: Introductionmentioning

confidence: 88%

Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

Jabłońska

2015

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…The term "almost all" in the above theorem has therefore to be understood in terms of prevalence [27]. As we are dealing with finite dimensional spaces of linear maps in this paper, we will from now on use the term "almost surly" to mean that the complement will have…”

Section: A Embedding Of Low Dimensional Compact Setsmentioning

confidence: 99%

Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces

Blumensath¹,

Davies²

2009

IEEE Trans. Inform. Theory

246

361

View full text Add to dashboard Cite

Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal basis. In fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies below the Nyquist rate. In this paper we consider a more general signal model and assume signals that live on or close to the union of linear subspaces of low dimension. We present sampling theorems for this model that are in the same spirit as the Nyquist-Shannon sampling theorem in that they connect the number of required samples to certain model parameters.Contrary to the Nyquist-Shannon sampling theorem, which gives a necessary and sufficient condition for the number of required samples as well as a simple linear algorithm for signal reconstruction, the model studied here is more complex. We therefore concentrate on two aspects of the signal model, the existence of one to one maps to lower dimensional observation spaces and the smoothness of the inverse map. We show that almost all linear maps are one to one when the observation space is at least of the same dimension as the largest dimension of the convex hull of the union of any two subspaces in the model. However, we also show that in order for the inverse map to have certain smoothness properties such as a given finite Lipschitz constant, the required observation dimension necessarily depends logarithmically This is a corrected version of the papers in which a few small errors have been corrected. Importantly, the dependence on δ in Theorem 3.3 and its corollaries has been corrected. VERSION: DECEMBER 3, 2009 1 on the number of subspaces in the signal model. In other words, whilst unique linear sampling schemes require a small number of samples depending only on the dimension of the subspaces involved, in order to have stable sampling methods, the number of samples depends necessarily logarithmically on the number of subspaces in the model. These results are then applied to two examples, the standard compressed sensing signal model in which the signal has a sparse representation in an orthonormal basis and to a sparse signal model with additional tree structure.

show abstract

“…(We note here that their result is in fact slightly stronger, giving a "prevalent" set of linear maps with this Hölder property, see [32,61]. )…”

Section: An Embedding Theoremmentioning

confidence: 83%

Parametrising the attractor of the two-dimensional Navier–Stokes equations with a finite number of nodal values

Friz

Robinson

2001

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We consider the solutions lying on the global attractor of the two-dimensional Navier-Stokes equations with periodic boundary conditions and analytic forcing. We show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely, proving a conjecture due to Foias and Temam. Our results also hold for the complex Ginzburg-Landau equation, the Kuramoto-Sivashinsky equation, and reaction-diffusion equations with analytic nonlinearities.

show abstract

Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces

Cited by 364 publications

References 30 publications

Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

Some analogies between Haar meager sets and Haar null sets in abelian Polish groups

Sampling Theorems for Signals From the Union of Finite-Dimensional Linear Subspaces

Parametrising the attractor of the two-dimensional Navier–Stokes equations with a finite number of nodal values

Contact Info

Product

Resources

About