On the Riesz energy of measures. (English) Zbl 1083.31008
Summary: Representations for the Riesz kernel \(|x-y|^{-s}\) \((s>0)\) are presented, which lead to new interpretations of the energy of measures. It is shown that the surface measure on the unit sphere in \(\mathbb R^3\) solves a minimal energy problem independent of \(s\) (but intimately related to Riesz \(s\)-energy) and that \(n\) points on the unit circle with minimal discrete Riesz energy are \(n\)th roots of unity, unique up to rotation. Moreover, the energy of signed measures is estimated in terms of their discrepancy.
MSC:
| 31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |
| 28A75 | Length, area, volume, other geometric measure theory |
References:
| [1] | Alexander, R.; Stolarsky, K. B., Extremal problems of distance geometry related to energy integrals, Trans. Amer. Math. Soc., 193, 1-31 (1974) · Zbl 0293.52005 |
| [2] | Andrievskii, V. V.; Blatt, H.-P., Discrepancy of Signed Measures and Polynomial Approximation (2001), Springer: Springer Heidelberg · Zbl 1013.42017 |
| [3] | Cui, J.; Freeden, W., Equidistribution on the sphere, SIAM J. Sci. Comput., 18, 595-609 (1997) · Zbl 0986.11049 |
| [4] | S.B. Damelin, P.J. Grabner, Energy functionals, numerical integration and asymptotic equidistribution on the sphere, J. Complexity, available online 10 December 2002.; S.B. Damelin, P.J. Grabner, Energy functionals, numerical integration and asymptotic equidistribution on the sphere, J. Complexity, available online 10 December 2002. · Zbl 1043.65049 |
| [5] | Drmota, M.; Tichy, R. F., Sequences, Discrepancies and Applications, Lecture Notes in Mathematics, Vol. 1651 (1997), Springer: Springer Berlin · Zbl 0877.11043 |
| [6] | Fefferman, C.; De la Llave, R., Relativistic stability of matter—I, Rev. Mat. Iberoamericana, 2, 119-160 (1986) · Zbl 0602.58015 |
| [7] | Götz, M., On the distribution of weighted extremal points on a surface in \(R^d, d\)⩾ 3, Potential Anal., 13, 345-359 (2000) · Zbl 0968.31006 |
| [8] | Götz, M., Potential theory and discrepancy estimates in higher dimensions, (Jain, P. K.; etal., Wavelets and Allied Topics (2001), Narosa: Narosa Delhi), 329-360 · Zbl 1038.42026 |
| [9] | Götz, M., Discrepancy and the error in integration, Monatsh. Math., 136, 99-121 (2002) · Zbl 1018.11039 |
| [10] | Götz, M.; Saff, E. B., Note on \(d\)-extremal configurations for the sphere in \(R^{d+1}\), Int. Ser. Num. Math., 137, 159-162 (2001) · Zbl 0988.31004 |
| [11] | Grabner, P. J.; Tichy, R. F., Spherical designs, discrepancy and numerical integration, Math. Comp., 60, 327-336 (1993) · Zbl 0795.65011 |
| [12] | Hüsing, J., Abschätzungen für die Diskrepanz von signierten Maßen durch deren Energienorm, Dissertation (1998), Shaker-Verlag: Shaker-Verlag Aachen · Zbl 0948.30045 |
| [13] | Hüsing, J., Estimates for the discrepancy of a signed measure using its energy norm, J. Approx. Theory, 109, 1-29 (2001) · Zbl 0976.30009 |
| [14] | Kleiner, W., Une condition de Dini-Lipschitz dans la théorie du potentiel, Ann. Polon. Math., XIV, 117-130 (1964) · Zbl 0131.10203 |
| [15] | Kleiner, W., On the equilibrium of signed measures, Colloq. Math., XI, 258-278 (1964) · Zbl 0141.30502 |
| [16] | Kuijlaars, A. B.J.; Saff, E. B., Asymptotics for minimal discrete energy on the sphere, Trans. Amer. Math. Soc., 350, 523-538 (1998) · Zbl 0896.52019 |
| [17] | Landkof, N. S., Foundations of Modern Potential Theory (1972), Springer: Springer Berlin · Zbl 0253.31001 |
| [18] | Saff, E. B.; Kuijlaars, A. B.J., Distributing many points on a sphere, Math. Intelligencer, 19, 5-11 (1997) · Zbl 0901.11028 |
| [19] | Smyth, C. J., Inequalities relating different definitions of discrepancy, J. Austral. Math. Soc. A, 17, 81-87 (1974) · Zbl 0276.10032 |
| [20] | Steinacker, J.; Thamm, E.; Maier, U., Efficient integration of intensity functions on the unit sphere, J. Quant. Spectrosc. Radiat. Transfer, 56, 97-107 (1996) |
| [21] | Stolarsky, K. B., Sums of distances between points on a sphere, Proc. Amer. Math. Soc., 35, 547-549 (1972) · Zbl 0255.52004 |
| [22] | Stolarsky, K. B., Sums of distances between points on a sphere II, Proc. Amer. Math. Soc., 41, 575-582 (1973) · Zbl 0274.52012 |
| [23] | Fejes Tóth, G.; Fejes Tóth, L., Dictators on a planet, Studia Sci. Math. Hung., 15, 313-316 (1980) · Zbl 0512.52007 |
| [24] | Wagner, G., On the product of distances to a point set on a sphere, J. Austral. Math. Soc. A, 47, 466-482 (1989) · Zbl 0706.11046 |
| [25] | Wagner, G., On means of distances on the surface of a sphere (lower bounds), Pacific J. Math., 144, 389-398 (1990) · Zbl 0648.10033 |
| [26] | Wagner, G., On means of distances on the surface of a sphere II (upper bounds), Pacific J. Math., 154, 381-396 (1992) · Zbl 0767.11033 |
| [27] | Y.M. Zhou, Equidistribution and Extremal Energy of \(N\); Y.M. Zhou, Equidistribution and Extremal Energy of \(N\) · Zbl 0932.58025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
