Beurling–Landauʼs density on compact manifolds

Ortega-Cerdà, Joaquim; Pridhnani, Bharti

doi:10.1016/j.jfa.2012.07.004

Cited by 19 publications

(20 citation statements)

References 13 publications

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“…We follow the approach of [21,31], which compare a discrete set of reproducing kernels to a continuous resolution of the identity in the space of band-limited functions. Other approaches, such as the original technique of Landau [26,27], the technique of Ortega-Cerdà and Pridhnani [32], or the approach of [14,16], might be successful as well, but these will require additional features, such as the existence of a Riesz basis of reproducing kernels.…”

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confidence: 99%

What Is Variable Bandwidth?

Gröchenig

Klotz

2017

Comm Pure Appl Math

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Abstract. We propose a new notion of variable bandwidth that is based on the spectral subspaces of an elliptic operatorf where p > 0 is a strictly positive function. Denote by c Λ (A p ) the orthogonal projection of A p corresponding to the spectrum of A p in Λ ⊂ R + , the range of this projection is the space of functions of variable bandwidth with spectral set in Λ.We will develop the basic theory of these function spaces. First, we derive (nonuniform) sampling theorems, second, we prove necessary density conditions in the style of Landau. Roughly, for a spectrum Λ = [0, Ω] the main results say that, in a neighborhood of x ∈ R, a function of variable bandwidth behaves like a bandlimited function with local bandwidth (Ω/p(x)) 1/2 .Although the formulation of the results is deceptively similar to the corresponding results for classical bandlimited functions, the methods of proof are much more involved. On the one hand, we use the oscillation method from sampling theory and frame theoretic methods, on the other hand, we need the precise spectral theory of Sturm-Liouville operators and the scattering theory of one-dimensional Schrödinger operators.

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confidence: 99%

What Is Variable Bandwidth?

Gröchenig

Klotz

2017

Comm Pure Appl Math

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“…The next result provides us with Riesz sequences of reproducing kernels with cardinality almost optimal. See [9] for the definition of separation in the complex setting, and [9,13] for a proof of the following result in the two different settings we are considering. Remark 2.9.…”

Section: Resultsmentioning

confidence: 99%

“…This is the only point in the real setting that one uses the hypothesis that M is admissible. In [13], the admissibility is needed to establish the connection between Fekete points and interpolating arrays. The equivalent version of Theorem 2.8 that we need is the following: if Z(L) a set of Fekete points of degree L for M, then Z(L) is interpolating for the…”

Section: Resultsmentioning

confidence: 99%

Uniformly bounded orthonormal polynomials on the sphere

Marzo¹,

Ortega-Cerdà²

2015

Bull. London Math. Soc.

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ABSTRACT. Given any ε > 0, we construct an orthonormal system of n k uniformly bounded polynomials of degree at most k on the unit sphere in R m+1 where n k is bigger than 1 − ε times the dimension of the space of polynomials of degree at most k. Similarly we construct an orthonormal system of sections of powers L k of a positive holomorphic line bundle on a compact Kähler manifold with cardinality bigger than 1 − ε times the dimension of the space of global holomorphic sections to L k .

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“…There are many interesting results related to the problem of the equidistribution of Fekete points (see [1,12,17]). …”

Section: Proof Of Theorem 17mentioning

confidence: 99%

“…Lyubarskii and K. Seip [19,10], in the case of the PaleyWiener space. In high dimensional case, J. Marzo found in [11] necessary density conditions for L p -MZ families and L p -interpolating families for 1 ≤ p ≤ ∞ on the sphere, and J. Ortega-Cerdà and B. Pridhnani gave in [17] necessary density conditions for L 2 -MZ families and L 2 -interpolating families on a smooth, connected, compact Riemannian manifold without boundary. This result can be seen as the analogue of the Paley-Wiener space result due to H.J.…”

Section: Introductionmentioning

confidence: 99%