Quadrature in Besov spaces on the Euclidean sphere. (English) Zbl 1134.41013
Let \({\mathbb S}^q\subset{\mathbb R}^{q+1}\) be the unit sphere and \(\mu_q\) be its Lebesgue surface measure. To approximate the surface integrals \(\int_{{\mathbb S}^q}fd\mu_q\) the quadratures of the form \(\sum_{k=1}^Mw_kf(x_k)\), where \(x_k\in{\mathbb S}^q\), \(w_k\in{\mathbb R}\), \(k=1,\dots,M\), are considered for functions \(f:{\mathbb S}^q\to{\mathbb R}\) belonging to the Besov space on \({\mathbb S}^q\) (i.e. some subset of \(L^p\)-space on \({\mathbb S}^q\) equipped with a quasi-norm). Upper and lower error bounds are proved under the usual assumption that the involved quadratures are exact for spherical polynomials of a given degree. The complexity of such quadratures is also established.
Reviewer: Szymon Wąsowicz (Bielsko-Biała)
MSC:
| 41A55 | Approximate quadratures |
| 65D30 | Numerical integration |
| 65D32 | Numerical quadrature and cubature formulas |
Keywords:
Besov spaces on the sphere; numerical integration; quadrature formulas; spherical polynomialsReferences:
| [1] | Brauchart, J. S.; Hesse, K., Numerical integration over spheres of arbitrary dimension, Constr. Approx., 25, 41-71 (2007) · Zbl 1106.41028 |
| [2] | Brown, G.; Dai, F.; Sun, Y. S., Kolmogorov widths of classes of smooth functions on the sphere \(S^{d - 1}\), J. Complexity, 18, 1001-1023 (2002) · Zbl 1036.41012 |
| [3] | Dai, F., Jackson-type inequality for doubling weights on the sphere, Constr. Approx., 26, 1, 91-112 (2006) · Zbl 1116.41010 |
| [4] | Dai, F., Characterizations of function spaces on the sphere using frames, Trans. Amer. Math. Soc., 359, 2, 567-589 (2007) · Zbl 1184.42024 |
| [5] | DeVore, R. A.; Lorentz, G. G., Constructive Approximation (1993), Springer: Springer Berlin · Zbl 0797.41016 |
| [6] | Driscoll, J. R.; Healy, D. M., Computing Fourier transforms and convolutions on the 2-sphere, Adv. Appl. Math., 15, 202-250 (1994) · Zbl 0801.65141 |
| [7] | A. Erdélyi (Ed.), W. Magnus, F. Oberhettinger, F. G. Tricomi (research associates), Higher Transcendental Functions, vol. II, California Institute of Technology, Bateman Manuscript Project, McGraw-Hill Book Company, Inc., New York, Toronto, London, 1953.; A. Erdélyi (Ed.), W. Magnus, F. Oberhettinger, F. G. Tricomi (research associates), Higher Transcendental Functions, vol. II, California Institute of Technology, Bateman Manuscript Project, McGraw-Hill Book Company, Inc., New York, Toronto, London, 1953. · Zbl 0052.29502 |
| [8] | Le Gia, Q. T.; Mhaskar, H. N., Polynomial operators and local approximation of solutions of pseudo-differential equations on the sphere, Numer. Math., 103, 299-322 (2006) · Zbl 1090.65133 |
| [9] | Hesse, K., A lower bound for the worst-case cubature error on spheres of arbitrary dimension, Numer. Math., 103, 413-433 (2006) · Zbl 1089.41025 |
| [10] | Hesse, K.; Sloan, I. H., Cubature over the sphere \(S^2\) in Sobolev spaces of arbitrary order, J. Approx. Theory., 141, 118-133 (2006) · Zbl 1102.41027 |
| [11] | Hesse, K.; Sloan, I. H., Optimal lower bounds for cubature error on the sphere \(S^2\), J. Complexity, 21, 790-803 (2005) · Zbl 1099.41023 |
| [12] | Jetter, K.; Stöckler, J.; Ward, J. D., Norming sets and spherical cubature formulas, (Chen, Z.; Li, Y.; Micchelli, C.; Xu, Y., Computational Mathematics (1998), Marcel Decker: Marcel Decker New York), 237-245 · Zbl 0931.65013 |
| [13] | Lizorkin, P. I.; Rustamov, Kh. P., Nikolskii-Besov spaces on the sphere in connection with approximation theory, Tr. Mat. Inst. Steklova, 204, 172-201 (1993), (Proc. Steklov Inst. Math. 3 (1994), 149-172) · Zbl 0849.46025 |
| [14] | Mhaskar, H. N., Polynomial operators and local smoothness classes on the unit interval, J. Approx. Theory, 131, 243-267 (2004) · Zbl 1083.41004 |
| [15] | Mhaskar, H. N., On the representation of smooth functions on the sphere using finitely many bits, Appl. Comput. Harmon. Anal., 18, 3, 215-233 (2005) · Zbl 1082.41020 |
| [16] | Mhaskar, H. N., Weighted quadrature formulas and approximation by zonal function networks on the sphere, J. Complexity, 22, 348-370 (2006) · Zbl 1103.65028 |
| [17] | Mhaskar, H. N.; Narcowich, F. J.; Ward, J. D., Approximation properties of zonal function networks using scattered data on the sphere, Adv. Comput. Math., 11, 121-137 (1999) · Zbl 0939.41012 |
| [18] | Mhaskar, H. N.; Narcowich, F. J.; Ward, J. D., Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature, Math. Comput., 70, 235, 1113-1130 (2001), Corrigendum: Math. Comput. 71 (2001) 453-454 · Zbl 0980.76070 |
| [19] | Mhaskar, H. N.; Prestin, J., Polynomial frames: a fast tour, (Chui, C. K.; Neamtu, M.; Schumaker, L. L., Approximation Theory, XI, Gatlinburg, 2004 (2005), Nashboro Press: Nashboro Press Brentwood), 287-318 · Zbl 1077.42028 |
| [20] | C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, vol. 17, Springer, Berlin, 1966.; C. Müller, Spherical Harmonics, Lecture Notes in Mathematics, vol. 17, Springer, Berlin, 1966. · Zbl 0138.05101 |
| [21] | Pawelke, S., Über die Approximationsordnung bei Kugelfunktionen und algebraischen Polynomen, Tôhoku Math. J., 24, 473-486 (1972) · Zbl 0243.41006 |
| [22] | Potts, D.; Steidl, G.; Tasche, M., Fast algorithms for discrete polynomial transforms, Math. Comput., 67, 1577-1590 (1998) · Zbl 0904.65145 |
| [23] | Ragozin, D. L., Constructive polynomial approximation on sphere and projective spaces, Trans. Amer. Math. Soc., 162, 157-170 (1971) · Zbl 0234.41011 |
| [24] | Reimer, M., Hyperinterpolation on the sphere at the minimal projection order, J. Approx. Theory, 104, 2, 272-286 (2000) · Zbl 0959.41001 |
| [25] | Sloan, I. H.; Womersley, R. S., Extremal systems of points and numerical integration on the sphere, Adv. Comput. Math., 21, 107-125 (2004) · Zbl 1055.65038 |
| [26] | Stein, E. M.; Weiss, G., Fourier Analysis on Euclidean Spaces (1971), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0232.42007 |
| [27] | Stroud, A. H., Approximate Calculation of Multiple Integrals (1971), Prentice-Hall: Prentice-Hall Inc., Englewood Cliffs, NJ · Zbl 0379.65013 |
| [28] | G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloqium Publication, vol. 23, American Mathematical Society, Providence, RI, 1975.; G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloqium Publication, vol. 23, American Mathematical Society, Providence, RI, 1975. · Zbl 0305.42011 |
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.
