Distribution of point charges with small discrete energy

Pritsker, Igor E.

doi:10.1090/s0002-9939-2011-11135-3

Cited by 8 publications

(7 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also refer to Baglama et al . [6], Lopez‐Garcia and Saff [33], Pritsker [36, 37], and [5, 26 41, 47, 48].…”

Section: Statement Of Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Polynomials With Zeros on the Unit Circle: Regularity of Leja Sequences

Steinerberger

2021

Mathematika

View full text Add to dashboard Cite

Let z 1 , . . . , z m be m distinct complex numbers, normalized to |z k | = 1, and consider the polynomialWe define a sequence of polynomials in a greedy fashion,whereand prove that, independently of the initial polynomial p m , the roots of p N equidistribute in angle at rate at most (log N ) 2 /N. This persists when sometimes adding "adversarial" points by hand. We also obtain sharp rates for an L 2 -version of a problem first raised by Erdős and solved by Beck in L ∞ . §1. Introduction.1.1. Introduction. Let (x n ) ∞ n=1 be a sequence on [0,1]. We define the discrepancy function D N : [0, 1] → [0, 1] associated of the first N elements via

show abstract

“…We also refer to Baglama et al . [6], Lopez‐Garcia and Saff [33], Pritsker [36, 37], and [5, 26 41, 47, 48].…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…A philosophically related object was studied in [2,27,28]. We also refer to Baglama et al [6], Lopez-Garcia and Saff [33], Pritsker [36,37], and [5, 26 41, 47, 48]. §3.…”

Section: At the Same This Mysterious Function Log (2| Sin (πX)|) Arises Naturally As An Infinite Fourier Seriesmentioning

confidence: 99%

Polynomials With Zeros on the Unit Circle: Regularity of Leja Sequences

Steinerberger

2021

Mathematika

View full text Add to dashboard Cite

show abstract

“…A philosophically related object was studied in [2,30,31]. We also refer to Baglama, Calvetti & Reichel [6], Lopez-Garcia & Saff [36], Pritsker [39,40] and [5,29,44,50,51].…”

Section: At the Same This Mysterious Function Log (2 |Sin (πX)|) Aris...mentioning

confidence: 99%

Polynomials with Zeros on the Unit Circle: Regularity of Leja Sequences

Steinerberger¹

2020

Preprint

View full text Add to dashboard Cite

Let z 1 , . . . , zm be m distinct complex numbers, normalized to |z k | = 1, and consider the polynomialWe define a sequence of polynomials in a greedy fashion,where z * = arg maxand prove that, independently of the initial polynomial pm, the roots of p N equidistribute in angle at rate at most (log N ) 2 /N . This even persists when sometimes adding 'adversarial' points by hand. We rephrase the main result in terms of a dynamical system involving the inverse fractional Laplacian (−∆) −1/2 and conjecture that, when phrased in this language, the underlying regularity phenomenon might appear in a very general setting.

show abstract

“…In recent years there has been an increasing interest in the study of Leja and greedy energy sequences in Euclidean spaces, as they provide many interesting applications for different areas in mathematics. In addition to the references mentioned above, we find diverse applications in works on numerical analysis, numerical linear algebra, approximation theory [1,4,6,7,8,9,10,15,26,30], potential theory [9,13,17,24,27], discrepancy theory [23,29], stochastic analysis [22], among other areas.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of the minimum values of Riesz and logarithmic potentials generated by greedy energy sequences on the unit circle

López-García¹,

McCleary²

2021

Preprint

View full text Add to dashboard Cite

In this paper we investigate greedy energy sequences on the unit circle for the logarithmic and Riesz potentials. By definition, if (an) ∞ n=0 is a greedy s-energy sequence on the unit circle, the Riesz potential UN,s(x) := N−1 k=0 |a k − x| −s , s > 0, generated by the first N points of the sequence attains its minimum value on the circle at the point aN . In this work, we analyze the asymptotic properties of these extremal values UN,s(aN ), treating separately the cases 0 < s < 1, s = 1, and s > 1. We present new second-order asymptotic formulas for UN,s(aN ) in the cases 0 < s < 1 and s = 1. A first-order result for s > 1 is proved, and it is shown that the first-order normalized sequence (UN,s(aN )/N s ) ∞ N=1 is divergent in this case.

show abstract

Distribution of point charges with small discrete energy

Cited by 8 publications

References 19 publications

Polynomials With Zeros on the Unit Circle: Regularity of Leja Sequences

Polynomials With Zeros on the Unit Circle: Regularity of Leja Sequences

Polynomials with Zeros on the Unit Circle: Regularity of Leja Sequences

Asymptotics of the minimum values of Riesz and logarithmic potentials generated by greedy energy sequences on the unit circle

Contact Info

Product

Resources

About