Fekete Potentials and Polynomials for Continua

Korevaar, J.; Monterie, M.A.

doi:10.1006/jath.2000.3535

Cited by 6 publications

(5 citation statements)

References 20 publications

(16 reference statements)

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“…The symmetry breaking phenomenon for the configurations minimizing energy (2.1) on certain planar curves was studied in [4]. Papers [24] and [25] estimate the difference between the potential of the equilibrium distribution on closed smooth Jordan arcs on the plane and the potential of the minimum Riesz energy configurations for s = 0. The complete asymptotic expansion for the minimal Riesz s-energy of N equally spaced points on the circle in terms of powers of N was found in [9] and, for the Riemannian circle, in [8].…”

Section: Review Of Known Resultsmentioning

confidence: 99%

Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

Borodachov¹

2012

Can. j. math.

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We consider the problem of minimizing the energy of N points repelling each other on curves in ℝ ∞d with the potential |x — y|—s, s ≥ 1, where |・| is the Euclidean norm. For a sufficiently smooth, simple, closed, regular curve, we find the next order term in the asymptotics of the minimal s-energy. On our way, we also prove that at least for s ≥ 2, the minimal pairwise distance in optimal configurations asymptotically equals L/N, N → 1, where L is the length of the curve.

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Section: Review Of Known Resultsmentioning

confidence: 99%

Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

Borodachov¹

2012

Can. j. math.

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“…It is known that Leja points are equidistributed in E. Theorem 2.2 provides new quantitative information about discrete potentials of Fekete and Leja points for non-smooth sets. Surveys of results on Fekete points may be found in Korevaar [14], Andrievskii and Blatt [2] and Korevaar and Monterie [15]. We note that the estimates of Theorem 2.2 can be improved for the Fekete points of a set E satisfying more restrictive smoothness conditions.…”

Section: )mentioning

confidence: 83%

“…Surveys of results on Fekete points may be found in Korevaar [14], Andrievskii and Blatt [2] and Korevaar and Monterie [15]. We note that the estimates of Theorem 2.2 can be improved for the Fekete points of a set E satisfying more restrictive smoothness conditions.…”

Section: Rate Of Convergence and Discrepancy In Equidistributionmentioning

confidence: 83%

Distribution of point charges with small discrete energy

Pritsker

2011

Proc. Amer. Math. Soc.

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Abstract. We study the asymptotic equidistribution of points near arbitrary compact sets of positive capacity in R d , d ≥ 2. Our main tools are the energy estimates for Riesz potentials. We also consider the quantitative aspects of this equidistribution in the classical Newtonian case. In particular, we quantify the weak convergence of discrete measures to the equilibrium measure and give the estimates of convergence rates for discrete potentials to the equilibrium potential. Asymptotic equidistribution of discrete setsWe consider the potential theory associated with Riesz kernelsFor a Borel measure μ with compact support, define its energy by dμ(x)dμ(y).A central theme in potential theory is the study of the minimum energy problemwhere M(E) is the space of all positive unit Borel measures supported on E. If Robin's constant W α (E) is finite, then the above infimum is attained by the equilibrium measure μ E ∈ M(E) [16, pp. 131-133], which is a unique probability measure expressing the steady state distribution of charge on the conductor E. The capacity of E is defined bywhere we set C α (E) = 0 when W α (E) is infinite. For a more detailed exposition of Riesz potential theory, we refer the reader to the book of Landkof [16].

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“…Thus Theorem 2.2 provides new information about Leja points and corresponding polynomials for quite general sets. Surveys of results on Fekete points and Fekete polynomials may be found in Korevaar [23], Andrievskii and Blatt [4] and Korevaar and Monterie [24]. We note that the estimates of Theorem 2.2 can be improved for the Fekete points and Fekete polynomials of a set E satisfying more restrictive smoothness conditions.…”

Section: Rate Of Convergence and Discrepancy In Equidistributionmentioning

confidence: 92%

Equidistribution of points via energy

Pritsker

2011

Ark. Mat.

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We study the asymptotic equidistribution of points with discrete energy close to Robin's constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates of growth for the Fekete and Leja polynomials associated with large classes of compact sets, convergence rates of the discrete energy approximations to Robin's constant, and problems on the means of zeros of polynomials with integer coefficients.

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Fekete Potentials and Polynomials for Continua

Cited by 6 publications

References 20 publications

Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves

Distribution of point charges with small discrete energy

Equidistribution of points via energy

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