The Diamond ensemble: A constructive set of spherical points with small logarithmic energy

Beltrán, Carlos; Etayo, Ujué

doi:10.1016/j.jco.2020.101471

Cited by 20 publications

(13 citation statements)

References 16 publications

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“…In general, at best only one or two terms are known; cf. [6,12,22,35,36] in case of the sphere and [14,38] for curves. A notable exception are the minimal energy asymptotics for the unit circle for a whole class of energy functionals for which equally spaced points are optimal configurations.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Complete minimal logarithmic energy asymptotics for points in a compact interval: a consequence of the discriminant of Jacobi polynomials

Brauchart

2021

Preprint

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The electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as n → ∞ of the minimal logarithmic potential energy of n point charges restricted to move in the interval [−1, 1] in the presence of an external field generated by endpoint charges. By the same methods, we determine the complete asymptotic expansion of the logarithmic energy j =k log(1/|x j − x k |) of Fekete points, which, by definition, maximize the product of all mutual distances j =kThe results for other compact intervals differ only in the quadratic and linear term of the asymptotics. Explicit formulas and their asymptotics follow from the discriminant, leading coefficient, and special values at ±1 of Jacobi polynomials. For all these quantities we derive complete Poincaré-type asymptotics.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Complete minimal logarithmic energy asymptotics for points in a compact interval: a consequence of the discriminant of Jacobi polynomials

Brauchart

2021

Preprint

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“…Using the three-term relation in (5) and orthogonality in ( 6), the following lemma can be easily obtained.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…The p-frame potentials of DPPs will be studied in future. Although only three typical DPPs are addressed herein, various point processes on the sphere were recently proposed by Beltrán and Etayo [3,4,5], which will be addressed in future.…”

Section: Introductionmentioning

confidence: 99%

Finite frames, frame potentials and determinantal point processes on the sphere

Hirao

2021

Preprint

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Herein, we address the expectations of frame potentials of three types of determinantal point processes (DPPs) on the unit sphere S d : (i) spherical ensembles on S 2 ; (ii) harmonic ensembles on S d and (iii) jittered sampling point processes on S d . The random point configurations generated by such DPPs converge more rapidly towards finite unit norm tight frames (FUNTFs) than the Poisson point processes on the unit sphere.

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“…The images of these points through the stereographic projection are the zeros of the polynomials in Theorem 1.5. The set P N will be a union of equidistributed points in symmetric parallels with respect to the xy plane and the construction is similar to the one in [2].…”

Section: Construction Of the Point Set P Nmentioning

confidence: 99%

“…respectively with the Bombieri-Weyl norm and the Fubini-Study metric. In [17] it was proved that (2) µ(g, ζ) = g ζ N −1 |(Dg(ζ) | ζ ⊥ ) −1 |, (the definition and theory in [17] applies to the more general case of polynomial systems). Here, Dg(ζ) is just the derivative Dg(ζ) = ∂ ∂x g(x, y) ∂ ∂y g(x, y) (x,y)=ζ…”

Section: Introductionmentioning

confidence: 99%