# Minimum Norm (MN) Interpolation points on the Sphere S2

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Summary of data for Minimum Norm (MN) points.

 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

• Minimum norm (MN) points, Degree n = 1, Number of points dn = 4
•  MN points n = 1, dn = 4 Mesh norm function, h = 1.231 Interpolation norm function, ||Ln|| = 2 Eigenvalues of G lmin = 0.318, lavg = 0.318, lmax = 0.318
• Minimum norm (MN) points, Degree n = 2, Number of points dn = 9
•  MN points,  n = 2, dn = 9 Mesh norm function, h = 0.848 Interpolation norm function ||Ln|| = 3.06 Eigenvalues of G lmin = 0.387, lavg = 0.716, lmax = 0.978
• Minimum norm (MN) points, Degree n = 3, Number of points dn = 16
•  MN points, n = 3, dn = 16 Mesh norm function, h = 0.595 Interpolation norm function, ||Ln|| = 3.56 Eigenvalues of G lmin = 0.825, lavg = 1.273, lmax = 2.278
• Minimum norm (MN) points, Degree n = 4, Number of points dn = 25
•  MN points n = 4, dn = 25 Mesh norm function h = 0.500 Interpolation norm function ||Ln|| = 4.53 Eigenvalues of G lmin = 0.856, lavg = 1.989, lmax = 3.293
• Minimum norm (MN) points, Degree n = 5, Number of points dn = 36
•  MN points n = 5, dn = 36 Mesh norm function h = 0.417 Interpolation norm function ||Ln|| = 5.23 Eigenvalues of G lmin = 1.175, lavg = 2.865, lmax = 4.704
• Minimum norm (MN) points, Degree n = 6, Number of points dn = 49
•  MN points n = 6, dn = 49 Mesh norm function h = 0.366 Interpolation norm function ||Ln|| = 6.13 Eigenvalues of G lmin = 1.453, lavg = 3.899, lmax = 6.825
• Minimum norm (MN) points, Degree n = 7, Number of points dn = 64
•  MN points n = 7, dn = 64 Mesh norm function h = 0.305 Interpolation norm function ||Ln|| = 6.59 Eigenvalues of G lmin = 1.610, lavg = 5.093, lmax = 8.734
• Minimum norm (MN) points, Degree n = 8, Number of points dn = 81
•  MN points n = 8, dn = 81 Mesh norm function h = 0.285 Interpolation norm function ||Ln|| = 7.45 Eigenvalues of G lmin = 1.869, lavg = 6.446, lmax = 11.97
• Minimum norm (MN) points, Degree n = 9, Number of points dn = 100
•  MN points n = 9, dn = 100 Mesh norm function h =0.253 Interpolation norm function ||Ln|| = 8.18 Eigenvalues of G lmin = 2.112, lavg = 7.958, lmax = 14.66
• Minimum norm (MN) points, Degree n = 10, Number of points dn = 121
•  MN points n = 10, dn = 121 Mesh norm function h =0.231 Interpolation norm function ||Ln|| = 8.85 Eigenvalues of G lmin = 2.367, lavg = 9.629, lmax = 16.73
• Minimum norm (MN) points, Degree n = 11, Number of points dn = 144
•  MN points n = 11, dn = 144 Mesh norm function h = 0.218 Interpolation norm function ||Ln|| = 9.61 Eigenvalues of G lmin = 2.768, lavg = 11.46, lmax = 20.79
• Minimum norm (MN) points, Degree n = 12, Number of points dn = 169
•  MN points n = 12, dn = 169 Mesh norm function h = 0.204 Interpolation norm function ||Ln|| = 10.28 Eigenvalues of G lmin = 3.165, lavg = 13.45, lmax = 24.55
• Minimum norm (MN) points, Degree n = 13, Number of points dn = 196
•  MN points n = 13, dn = 196 Mesh norm function h = 0.187 Interpolation norm function ||Ln|| = 11.03 Eigenvalues of G lmin = 3.261, lavg = 15.60, lmax = 28.37
• Minimum norm (MN) points, Degree n = 14, Number of points dn = 225
•  MN points n = 14, dn = 225 Mesh norm function h = 0.178 Interpolation norm function ||Ln|| = 11.50 Eigenvalues of G lmin = 3.758, lavg = 17.90, lmax = 36.12
• Minimum norm (MN) points, Degree n = 15, Number of points dn = 256
•  MN points n = 15, dn = 256 Mesh norm function h = 0.171 Interpolation norm function ||Ln|| = 12.35 Eigenvalues of G lmin = 3.995, lavg = 20.37, lmax = 38.23
• Minimum norm (MN) points, Degree n = 16, Number of points dn = 289
•  MN points n = 16, dn = 289 Mesh norm function h = 0.162 Interpolation norm function ||Ln|| = 13.04 Eigenvalues of G lmin = 4.340, lavg = 23.000, lmax = 43.55
• Minimum norm (MN) points, Degree n = 17, Number of points dn = 324
•  MN points n = 17, dn = 324 Mesh norm function h = 0.153 Interpolation norm function ||Ln|| = 13.74 Eigenvalues of G lmin = 4.956, lavg = 25.78, lmax = 48.43
• Minimum norm (MN) points, Degree n = 18, Number of points dn = 361
•  MN points n = 18, dn = 361 Mesh norm function h = 0.146 Interpolation norm function ||Ln|| = 14.34 Eigenvalues of G lmin = 5.271, lavg = 28.73, lmax = 59.66
• Minimum norm (MN) points, Degree n = 19, Number of points dn = 400
•  MN points n = 19, dn = 400 Mesh norm function h = 0.140 Interpolation norm function ||Ln|| = 14.98 Eigenvalues of G lmin = 5.659, lavg = 31.83, lmax = 68.21
• Minimum norm (MN) points, Degree n = 20, Number of points dn = 441
•  MN points n = 20, dn = 441 Mesh norm function h = 0.134 Interpolation norm function ||Ln|| = 15.78 Eigenvalues of G lmin = 6.263, lavg = 35.09, lmax = 69.42
• Minimum norm (MN) points, Degree n = 21, Number of points dn = 484
•  MN points n = 21, dn = 484 Mesh norm function h = 0.126 Interpolation norm function ||Ln|| = 16.25 Eigenvalues of G lmin = 6.562, lavg = 38.52 , lmax = 76.78
• Minimum norm (MN) points, Degree n = 22, Number of points dn = 529
•  MN points n = 22, dn = 529 Mesh norm function h = 0.123 Interpolation norm function ||Ln|| = 17.08 Eigenvalues of G lmin = 6.867, lavg = 42.10, lmax = 84.21
• Minimum norm (MN) points, Degree n = 23, Number of points dn = 576
•  MN points n = 23, dn = 576 Mesh norm function h = 0.115 Interpolation norm function ||Ln|| = 17.62 Eigenvalues of G lmin = 7.323, lavg = 45.84, lmax = 87.44
• Minimum norm (MN) points, Degree n = 24, Number of points dn = 625
•  MN points n = 24, dn = 625 Mesh norm function h = 0.113 Interpolation norm function ||Ln|| = 18.50 Eigenvalues of G lmin = 7.566, lavg = 49.74, lmax = 106.45
• Minimum norm (MN) points, Degree n = 25, Number of points dn = 676
•  MN points n = 25, dn = 676 Mesh norm function h = 0.106 Interpolation norm function ||Ln|| = 18.72 Eigenvalues of G lmin = 7.625, lavg = 53.79, lmax = 103.28
• Minimum norm (MN) points, Degree n = 26, Number of points dn = 729
•  MN points n = 26, dn = 729 Mesh norm function h = 0.107 Interpolation norm function ||Ln|| = 19.73 Eigenvalues of G lmin = 7.420, lavg =58.01 , lmax = 118.27
• Minimum norm (MN) points, Degree n = 27, Number of points dn = 784
•  MN points n = 27, dn = 784 Mesh norm function h = 0.098 Interpolation norm function ||Ln|| = 20.42 Eigenvalues of G lmin = 8.780, lavg = 62.39, lmax = 121.42
• Minimum norm (MN) points, Degree n = 28, Number of points dn = 841
•  MN points n = 28, dn = 841 Mesh norm function h = 0.095 Interpolation norm function ||Ln|| = 21.12 Eigenvalues of G lmin = 8.223, lavg = 66.93, lmax = 136.24
• Minimum norm (MN) points, Degree n = 29, Number of points dn = 900
•  MN points n = 29, dn = 900 Mesh norm function h = 0.094 Interpolation norm function ||Ln|| = 21.40 Eigenvalues of G lmin = 9.31, lavg = 71.62, lmax = 145.35

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Rob Womersley, School of Mathematics, UNSW. Last updated 3-Nov-1999