Geodesic
Extremal properties of spherical semidesigns
Algebra i analiz, Tome 22 (2010) no. 5, pp. 131-139 Cet article a éte moissonné depuis la source Math-Net.Ru

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For every even $t\geq2$ and every set of vectors $\Phi=\{\varphi_1,\dots,\varphi_m\}$ on the sphere $S^{n-1}$, the notion of the $t$-potential $P_t(\Phi)=\sum^m_{i,j=1}[\langle\varphi_i,\varphi_j\rangle]^t$ is introduced. It is proved that the minimum value of the $t$-potential is attained at the spherical semidesigns of order $t$ and only at them. The first result of this type was obtained by B. B. Venkov. The result is extended to the case of sets $\Phi$ that do not lie on the sphere $S^{n-1}$. For the V. A. Yudin potentials $U_k(\Phi)$, $k=2,4,\dots,t$, it is shown that they attain the minimal value at the spherical semidesigns of order $t$ and only at them.
Keywords: spherical designs, spherical semidesigns. 
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     url = {http://geodesic.mathdoc.fr/item/AA_2010_22_5_a4/}
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N. O. Kotelina; A. B. Pevnyǐ. Extremal properties of spherical semidesigns. Algebra i analiz, Tome 22 (2010) no. 5, pp. 131-139. http://geodesic.mathdoc.fr/item/AA_2010_22_5_a4/

[1] Venkov B., “Réseaux et designs sphériques”, Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, Monogr. Enseign. Math., 37, Enseignement Math., Géneve, 2001, 10–86 | MR | Zbl

[2] Reznick B., Sums of even powers of real linear forms, Mem. Amer. Math. Soc., 96, no. 463, 1992, 155 pp. | MR

[3] Malozemov V. N., Pevnyi A. B., “Ravnougolnye zhestkie freimy”, Probl. mat. anal., 39, SPbGU, SPb., 2009, 3–25 | MR

[4] Delsarte P., Goethals J. M., Seidel J. J., “Spherical codes and designs”, Geom. Dedicata, 6 (1977), 363–388 | DOI | MR | Zbl

[5] Bannai E., Munemasa A., Venkov B., “The nonexistence of certain tight spherical designs”, Algebra i analiz, 16:4 (2004), 1–23 | MR | Zbl

[6] Haantjes J., “Equilateral point-sets in elliptic two- and three-dimensional spaces”, Nieuw. Arch. Wiskund (2), 22 (1948), 355–362 | MR | Zbl

[7] Benedetto J. J., Fickus M., “Finite normalized tight frames. Frame potentials”, Adv. Comput. Math., 18:2–4 (2003), 357–385 | DOI | MR | Zbl

[8] Casazza P. G., “Custom building finite frames”, Wavelets, Frames and Operator Theory, Contemp. Math., 345, Amer. Math. Soc., Providence, RI, 2004, 61–86 | MR | Zbl

[9] Sege G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[10] Sobolev S. L., Vvedenie v teoriyu kubaturnykh formul, Nauka, M., 1974 | MR

[11] Yudin V. A., “O variatsiyakh dizainov”, Probl. peredachi inf., 35:4 (1999), 68–73 | MR | Zbl

[12] Yudin V. A., “Vrascheniya sfericheskikh dizainov”, Probl. peredachi inf., 36:3 (2000), 39–45 | MR | Zbl

[13] Andreev N. N., Yudin V. A., “Ekstremalnye raspolozheniya tochek na sfere”, Mat. prosveschenie. Ser. 3, 1, 1997, 115–125