Abstract
A central problem in approximation theory is to characterize the best approximation of a function by polynomials, or other classes of simple functions, in terms of the smoothness of the function. In this chapter, we study the characterization of the best approximation by polynomials on the sphere. In the classical setting of one variable, the smoothness of a function on \({\mathbb{S}}^{1}\) is described by the modulus of smoothness, defined via the forward difference of the function.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Berens, H., Butzer, P.L., Pawelke, S.: Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten. Publ. Res. Inst. Math. Sci. Ser. A. 4, 201–268 (1968)
Calderoń, A.P., Weiss, G., Zygmund, A.: On the existence of singular integrals. In: Singular Integrals. Proceedings of Symposia in Pure Mathematics, Chicago, IL, pp. 56–73. American Mathematical Society, Providence (1966)
Dai, F., Ditzian, Z., Huang, H.W.: Equivalence of measures of smoothness in L p(S d − 1), 1 < p < ∞. Studia Math. 196, 179–205 (2010)
Dai, F., Xu, Y.: Moduli of smoothness and approximation on the unit sphere and the unit ball. Adv. Math. 224(4), 1233–1310 (2010)
Dai, F., Xu, Y.: Polynomial approximation in Sobolev spaces on the unit sphere and the unit ball. J. Approx. Theor. 163, 1400–1418 (2011)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, New York (1993)
Ditzian, Z.: A modulus of smoothness on the unit sphere. J. Anal. Math. 79, 189–200 (1999)
Ditzian, Z.: Jackson-type inequality on the sphere. Acta Math. Hungar. 102, 1–35 (2004)
Ditzian, Z.: Optimality of the range for which equivalence between certain measures of smoothness holds. Studia Math. 198, 271–277 (2010)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. McGraw-Hill, New York (1953)
Kamzolov, A.I.: The best approximation on the classes of functions W p α(S n) by polynomials in spherical harmonics. Mat. Zametki 32, 285–293 (1982). English translation in Math Notes, 32, 622–628 (1982)
Kurtz, D.S., Wheeden, R.L.: Results on weighted norm inequalities for multipliers. Trans. Am. Math. Soc. 255, 343–362 (1979)
Nikolskii, S.M., Lizorkin, P.I.: Approximation on the sphere, survey. In: Approximation and function spaces (Warsaw, 1986). Translated from the Russian by Jerzy Trzeciak, vol. 22, pp. 281–292. Banach Center Publication, PWN, Warsaw (1989)
Pawelke, S.: Über Approximationsordnung bei Kugelfunktionen und algebraischen Polynomen. Tôhoku Math. J. 24, 473–486 (1972)
Petrushev, P., Popov, V.: Rational Approximation of Real Functions. Cambridge University Press, Cambridge (1987)
Rudin, W.: Uniqueness theory for Laplace series. Trans. Am. Math. Soc. 68, 287–303 (1950)
Rustamov, Kh.P.: On the equivalence of some moduli of smoothness on the sphere. Soviet Math. Dokl. 44, 660–664 (1992)
Rustamov, Kh.P.: On the approximation of functions on a sphere (Russian). Izv. Ross. Akad. Nauk Ser. Mat. 57, 127–148 (1993). Translation in Russian Acad. Sci. Izv. Math. 43(2), 311–329 (1994)
Timan, A.F.: Theory of approximation of functions of a real variable. Translated from the Russian by J. Berry. Translation edited and with a preface by J. Cossar. Reprint of the 1963 English translation. Dover Publications, Mineola, New York (1994)
Wang, K.Y., Li, L.Q.: Harmonic Analysis and Approximation on the Unit Sphere. Science Press, Beijing (2000)
Zygmund, A.: Trigonometric Series, vols. I, II, 2nd edition. Cambridge University Press, New York (1959)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Dai, F., Xu, Y. (2013). Approximation on the Sphere. In: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6660-4_4
Download citation
DOI: https://doi.org/10.1007/978-1-4614-6660-4_4
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-6659-8
Online ISBN: 978-1-4614-6660-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)