Covering numbers of isotropic reproducing kernels on compact two‐point homogeneous spaces

Azevedo, Douglas; Barbosa, V. S.

doi:10.1002/mana.201600125

Cited by 12 publications

(9 citation statements)

References 34 publications

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“…Second, for selected values of α and β given by (2) with p and q given in Table 2, they are orthogonal over M d , as the following lemma describes, which is derived from the Funk-Hecke formula recently established in [3]. In the particular case M d = S d , the Funk-Hecke formula may be found in classical references such as [1,34].…”

Section: Orthogonality Properties Of Jacobi Polynomialsmentioning

confidence: 97%

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Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous Spaces

Malyarenko

2018

J Theor Probab

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A general form of the covariance matrix function is derived in this paper for a vector random field that is isotropic and mean square continuous on a compact connected two-point homogeneous space and stationary on a temporal domain. A series representation is presented for such a vector random field, which involve Jacobi polynomials and the distance defined on the compact two-point homogeneous space.

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Section: Orthogonality Properties Of Jacobi Polynomialsmentioning

confidence: 97%

“…This approach goes back to Cartan [10]. It has been used in both probabilistic literature [14] and approximation theory literature [3].…”

Section: An Approach Based On Lie Algebrasmentioning

confidence: 99%

Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous Spaces

Malyarenko

2018

J Theor Probab

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“…In this paper we present upper and lower estimates for the covering numbers of the embedding I W : H W → C(I) achieving tight bounds. The technique applied is similar to the one applied in [2] for the covering numbers of the embedding of the unit ball of the RKHS associated to isotropic kernels on compact two-point homogeneous space. In short, here, the approach is mainly based on operator norm estimate of I K and some others related finite rank operators.…”

Section: Introductionmentioning

confidence: 99%

Sharp estimates for the covering numbers of the Weierstrass fractal kernel

Azevedo¹,

González²,

Jordão³

2021

Preprint

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In this paper we use the infamous continuous and nowhere differentiable Weierstrass function as a prototype to define a "Weierstrass fractal kernel". We investigate properties of the reproducing kernel Hilbert space (RKHS) associated to this kernel by presenting an explicit characterization of this space. In particular, we show that this space has a dense subset composed of continuous but nowhere differentiable functions. Moreover, we present sharp estimates for the covering numbers of the unit ball of this space as a subset of the continuous functions.

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“…), and the Cayley elliptic plane P 16 (Cay) or P 16 (O). There are at least two different approaches to the subject of compact twopoint homogeneous spaces [21], including an approach based on Lie algebras and a geometric approach, which are used in probabilistic literature [2], [12], [22], statistical literaure [26], and approximation theory literature [3], [8]. All compact twopoint homogeneous spaces share the same property that all geodesics in a given one of these spaces are closed and have the same length [12].…”

Section: Introductionmentioning

confidence: 99%

Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces

Lü

2019

J Theor Probab

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The covariance matrix function is characterized in this paper for a Gaussian or elliptically contoured vector random field that is stationary, isotropic, and mean square continuous on the compact two-point homogeneous space. Necessary and sufficient conditions are derived for a symmetric and continuous matrix function to be an isotropic covariance matrix function on all compact two-point homogeneous spaces. It is also shown that, for a symmetric and continuous matrix function with compact support, if it makes an isotropic covariance matrix function in the Euclidean space, then it makes an isotropic covariance matrix function on the sphere or the real projective space.Keywords Covariance matrix function · Elliptically contoured random field · Gaussian random field · Isotropy · Stationarity · Jacobi polynomial · Bessel function Mathematics Subject Classification (2010) 60G60 · 62M10 · 62M30 IntroductionA d-dimensional compact two-point homogeneous space M d is a compact Riemannian symmetric space of rank one, and belongs to one of the following five families ([14], [29]): the unit spheres S d (d = 1, 2, . . .), the real projective spaces P d (R) (d = 2, 4, . . .), the complex projective spaces P d (C) (d = 4, 6, . . .), the quaternionic projective spaces P d (H) (d = 8, 12, . . .), and the Cayley elliptic plane P 16 (Cay) or

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Covering numbers of isotropic reproducing kernels on compact two‐point homogeneous spaces

Cited by 12 publications

References 34 publications

Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous Spaces

Time-Varying Isotropic Vector Random Fields on Compact Two-Point Homogeneous Spaces

Sharp estimates for the covering numbers of the Weierstrass fractal kernel

Isotropic Covariance Matrix Functions on Compact Two-Point Homogeneous Spaces

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