메뉴 건너뛰기




Volumn 77, Issue 263, 2008, Pages 1599-1613

Optimal logarithmic energy points on the unit sphere

Author keywords

Discrepancy; Fekete points; Logarithmic energy; Riesz energy; Sphere; Ultraspherical expansion

Indexed keywords


EID: 47249116205     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-08-02085-1     Document Type: Article
Times cited : (50)

References (32)
  • 2
    • 0000006032 scopus 로고    scopus 로고
    • V.V. Andrievskii, H.-P. Blatt, and M. Götz, Discrepancy estimates on the sphere, Monatsh. Math. 128 (1999), no. 3, 179-188. MR1719364 (2000h:11080)
    • V.V. Andrievskii, H.-P. Blatt, and M. Götz, Discrepancy estimates on the sphere, Monatsh. Math. 128 (1999), no. 3, 179-188. MR1719364 (2000h:11080)
  • 3
    • 84974189036 scopus 로고
    • Sums of distances between points on a sphere - an application of the theory of irregularities of distribution to discrete geometry
    • MR762175 86d:52004
    • J. Beck, Sums of distances between points on a sphere - an application of the theory of irregularities of distribution to discrete geometry, Mathematika 31 (1984), no. 1, 33-41. MR762175 (86d:52004)
    • (1984) Mathematika , vol.31 , Issue.1 , pp. 33-41
    • Beck, J.1
  • 4
    • 33745100763 scopus 로고    scopus 로고
    • J.S. Brauchart, About the second term of the asymptotics for optimal Riesz energy on the sphere in the potential-theoretical case, Integral Transforms Spec. Funct. 17 (2006), no. 5, 321-328. MR2237493 (2007f:31011)
    • J.S. Brauchart, About the second term of the asymptotics for optimal Riesz energy on the sphere in the potential-theoretical case, Integral Transforms Spec. Funct. 17 (2006), no. 5, 321-328. MR2237493 (2007f:31011)
  • 5
    • 33749644556 scopus 로고    scopus 로고
    • J.S. Brauchart and K. Hesse, Numerical integration over spheres of arbitrary dimension, Constr. Approx. 25 (2007), no. 1, 41-71. MR2263736 (2007f:41025)
    • J.S. Brauchart and K. Hesse, Numerical integration over spheres of arbitrary dimension, Constr. Approx. 25 (2007), no. 1, 41-71. MR2263736 (2007f:41025)
  • 6
    • 0038392442 scopus 로고    scopus 로고
    • S.B. Damelin and P.J. Grabner, Energy functionals, numerical integration and asymptotic equidistnbution on the sphere, J. Complexity 19 (2003), no. 3, 231-246. MR1984111 (2004e:41032)
    • S.B. Damelin and P.J. Grabner, Energy functionals, numerical integration and asymptotic equidistnbution on the sphere, J. Complexity 19 (2003), no. 3, 231-246. MR1984111 (2004e:41032)
  • 7
    • 47249109966 scopus 로고    scopus 로고
    • P.D. Dragnev, On the separation of logarithmic points on the sphere, Approximation theory, X (St. Louis, MO, 2001), Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2002, pp. 137-144. MR1924855 (2003h:41020)
    • P.D. Dragnev, On the separation of logarithmic points on the sphere, Approximation theory, X (St. Louis, MO, 2001), Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 2002, pp. 137-144. MR1924855 (2003h:41020)
  • 8
    • 0036972509 scopus 로고    scopus 로고
    • P.D. Dragnev, D.A. Legg, and D.W. Townsend, Discrete logarithmic energy on the sphere, Pacific J. Math. 207 (2002), no. 2, 345-358. MR1972249 (2004c:52015)
    • P.D. Dragnev, D.A. Legg, and D.W. Townsend, Discrete logarithmic energy on the sphere, Pacific J. Math. 207 (2002), no. 2, 345-358. MR1972249 (2004c:52015)
  • 9
    • 0034553901 scopus 로고    scopus 로고
    • d, d ≥ 3, Potential Anal. 13 (2000), no. 4, 345-359. MR1804177 (2002a:31006)
    • d, d ≥ 3, Potential Anal. 13 (2000), no. 4, 345-359. MR1804177 (2002a:31006)
  • 10
    • 0011265052 scopus 로고
    • Erdös-Turán type discrepancy bounds
    • MR1100852 92f:11108
    • P.J. Grabner, Erdös-Turán type discrepancy bounds, Monatsh. Math. 111 (1991), no. 2, 127-135. MR1100852 (92f:11108)
    • (1991) Monatsh. Math , vol.111 , Issue.2 , pp. 127-135
    • Grabner, P.J.1
  • 11
    • 27844497928 scopus 로고    scopus 로고
    • D.P. Hardin and E.B. Saff, Discretizing manifolds via minimum energy points, Notices Am. Math. Soc. 51 (2004), no. 10, 1186-1194. MR2104914 (2006a:41049)
    • D.P. Hardin and E.B. Saff, Discretizing manifolds via minimum energy points, Notices Am. Math. Soc. 51 (2004), no. 10, 1186-1194. MR2104914 (2006a:41049)
  • 12
    • 33646478548 scopus 로고    scopus 로고
    • K. Hesse, A lower bound for the worst-case cubature error on spheres of arbitrary dimension, Numer. Math. 103 (2006), no. 3, 413-433. MR2221056 (2007b:41050)
    • K. Hesse, A lower bound for the worst-case cubature error on spheres of arbitrary dimension, Numer. Math. 103 (2006), no. 3, 413-433. MR2221056 (2007b:41050)
  • 13
    • 27844500239 scopus 로고    scopus 로고
    • 2, J. Complexity 21 (2005), no. 6, 790-803. MR2182445 (2006m:65047)
    • 2, J. Complexity 21 (2005), no. 6, 790-803. MR2182445 (2006m:65047)
  • 14
    • 17044405176 scopus 로고    scopus 로고
    • 2, Bull. Austral. Math. Soc. 71 (2005), no. 1, 81-105. MR2127668 (2005k:41090)
    • 2, Bull. Austral. Math. Soc. 71 (2005), no. 1, 81-105. MR2127668 (2005k:41090)
  • 15
    • 33746576281 scopus 로고    scopus 로고
    • 2 in Sobolev spaces of arbitrary order
    • MR2252093
    • 2 in Sobolev spaces of arbitrary order, J. Approx. Theory 141 (2006), no. 2, 118-133. MR2252093
    • (2006) J. Approx. Theory , vol.141 , Issue.2 , pp. 118-133
    • Grabner, P.J.1
  • 16
    • 47249150659 scopus 로고    scopus 로고
    • J. Korevaar, Fekete extreme points and related problems, Approximation theory and function series (Budapest, 1995), Bolyai Soc. Math. Stud., 5, János Bolyai Math. Soc., Budapest, 1996, pp. 35-62. MR1432660 (98c:31004)
    • J. Korevaar, Fekete extreme points and related problems, Approximation theory and function series (Budapest, 1995), Bolyai Soc. Math. Stud., vol. 5, János Bolyai Math. Soc., Budapest, 1996, pp. 35-62. MR1432660 (98c:31004)
  • 17
    • 21944444965 scopus 로고    scopus 로고
    • Asymptotics for minimal discrete energy on the sphere
    • MR1458327 98e:11092
    • A.B.J. Kuijlaars and E.B. Saff, Asymptotics for minimal discrete energy on the sphere, Trans. Am. Math. Soc. 350 (1998), no. 2, 523-538. MR1458327 (98e:11092)
    • (1998) Trans. Am. Math. Soc , vol.350 , Issue.2 , pp. 523-538
    • Kuijlaars, A.B.J.1    Saff, E.B.2
  • 20
    • 33846959089 scopus 로고    scopus 로고
    • A partition of the unit sphere into regions of equal area and small diameter
    • electronic, MR2280380
    • P. Leopardi, A partition of the unit sphere into regions of equal area and small diameter, Electron. Trans. Numer. Anal. 25 (2006), 309-327 (electronic). MR2280380
    • (2006) Electron. Trans. Numer. Anal , vol.25 , pp. 309-327
    • Leopardi, P.1
  • 21
    • 0347174830 scopus 로고    scopus 로고
    • X.-J. Li and J.D. Vaaler, Some trigonometric extremal functions and Erdös-Turán type inequalities, Indiana Univ. Math. J. 48(1999), no. 1, 183-236. MR1722198 (2001a:11136)
    • X.-J. Li and J.D. Vaaler, Some trigonometric extremal functions and Erdös-Turán type inequalities, Indiana Univ. Math. J. 48(1999), no. 1, 183-236. MR1722198 (2001a:11136)
  • 25
    • 0004126854 scopus 로고
    • Chelsea Publishing Co, Bronx, New York, MR0393590 52:14399
    • E.D. Rainville, Special functions, Chelsea Publishing Co., Bronx, New York, 1971. MR0393590 (52:14399)
    • (1971) Special functions
    • Rainville, E.D.1
  • 26
    • 0001447029 scopus 로고
    • Minimal discrete energy on the sphere
    • MR1306011 96e:78011
    • E.A. Rakhmanov, E.B. Saff, and Y.M. Zhou, Minimal discrete energy on the sphere, Math. Res. Lett. 1(1994), no. 6, 647-662. MR1306011 (96e:78011)
    • (1994) Math. Res. Lett , vol.1 , Issue.6 , pp. 647-662
    • Rakhmanov, E.A.1    Saff, E.B.2    Zhou, Y.M.3
  • 27
    • 0031542184 scopus 로고    scopus 로고
    • Distributing many points on a sphere
    • MR1439152 98h:70011
    • E.B. Saff and A.B.J. Kuijlaars, Distributing many points on a sphere, Math. Intell. 19(1997), no. 1, 5-11. MR1439152 (98h:70011)
    • (1997) Math. Intell , vol.19 , Issue.1 , pp. 5-11
    • Saff, E.B.1    Kuijlaars, A.B.J.2
  • 28
    • 0002278399 scopus 로고
    • Estimates of mass distributions from their potentials and energies
    • MR0310268 46:9369
    • P. Sjogren, Estimates of mass distributions from their potentials and energies, Ark. Mat. 10(1972), 59-77. MR0310268 (46:9369)
    • (1972) Ark. Mat , vol.10 , pp. 59-77
    • Sjogren, P.1
  • 29
    • 0002587254 scopus 로고    scopus 로고
    • Mathematical problems for the next century
    • MR1631413 99h:01033
    • S. Smale, Mathematical problems for the next century, Math. Intelligencer 20(1998), no. 2, 7-15. MR1631413 (99h:01033)
    • (1998) Math. Intelligencer , vol.20 , Issue.2 , pp. 7-15
    • Smale, S.1
  • 30
    • 0035336208 scopus 로고    scopus 로고
    • W.J.H. Stortelder, J.J.B. de Swart, and J.D. Pintér, Finding elliptic Fekete points sets: two numerical solution approaches, J. Comput. Appl. Math. 130 (2001), no. 1-2, 205-216. MR1827981 (2002b:90137)
    • W.J.H. Stortelder, J.J.B. de Swart, and J.D. Pintér, Finding elliptic Fekete points sets: two numerical solution approaches, J. Comput. Appl. Math. 130 (2001), no. 1-2, 205-216. MR1827981 (2002b:90137)
  • 31
    • 84974288974 scopus 로고
    • On the product of distances to a point set on a sphere
    • MR1018975 90j:11080
    • G. Wagner, On the product of distances to a point set on a sphere, J. Aust. Math. Soc., Ser. A 47 (1989), no. 3, 466-482. MR1018975 (90j:11080)
    • (1989) J. Aust. Math. Soc., Ser. A , vol.47 , Issue.3 , pp. 466-482
    • Wagner, G.1
  • 32
    • 84973999972 scopus 로고
    • On means of distances on the surface of a sphere. II: Upper bounds
    • MR1159518 93b:52007
    • _, On means of distances on the surface of a sphere. II: Upper bounds, Pac J. Math. 154 (1992), no. 2, 381-396. MR1159518 (93b:52007)
    • (1992) Pac J. Math , vol.154 , Issue.2 , pp. 381-396
    • Wagner, G.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.