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Estimations of the remainder of spherical harmonic series

Published online by Cambridge University Press:  01 July 2008

XIRONG CHANG ,
FENG DAI  and
KUNYANG WANG
XIRONG CHANG
Affiliation:
Department of Mathematics and Physics, North China Electrical Power University, Beijing 102206, China. e-mail: [email protected]
FENG DAI
Affiliation:
Department of Mathematical and Statistical Sciences, CAB 632, University of Alberta, Edmonton, Alberta, CanadaT6G 2G1. e-mail: [email protected]
KUNYANG WANG*
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China. e-mail: [email protected]
*
‡Corresponding author, partially supported by NNSF of China under the grant # 10471010(2005–2007).
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Abstract

An asymptotic estimate is obtained for the error in approximation of functions by partial sums of spherical harmonic expansions. The precise constant in the main term is found and the order of growth of the remainder term is given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 243 - 255
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Andrews, G. E., Askey, R. and Roy, R.. Special functions. Encyclopedia of Mathematics and its Applications 71 (Cambridge University Press, 1999).CrossRefGoogle Scholar
[2]Askey, R. and Wainger, S.. On the behavior of special classes of ultraspherical expansions I. J. Analyse Math. 15 (1965), 193–220.CrossRefGoogle Scholar
[3]Brown, G. and Dai, F.. Approximation of smooth functions on compact two-point homogeneous spaces. J. Funct. Anal. 220 (2005), 401–423.CrossRefGoogle Scholar
[4]Brown, G., Dai, F. and Sun, Y. S.. Kolmogorov width of classes of smooth functions on the sphere . J. Complexity 18 (2002), no. 4, 1001–1023.Google Scholar
[5]Dunkl, C. F. and Xu, Yuan. Orthogonal Polynomials of Several Variables. (Cambridge University Press, 2001).CrossRefGoogle Scholar
[6]Kamzolov, A. I.. The best approximation of the classes of functions by polynomials in spherical harmonics. Math. Notes 32 (1982), 622–626.Google Scholar
[7]Kamzolov, A. I.. On the Kolmogorov diameters of classes of smooth functions on a sphere. Russian Math. Survey 44 (1989), no. 5, 196–197.CrossRefGoogle Scholar
[8]Kolmogorov, A. N.. Zur Grössenordnung des Restgliedes Fourierschen Reihen differenzierbarer Funktionen. Ann. of Math. (2) 36 (1935), 521–526.CrossRefGoogle Scholar
[9]Leontev, V. O.. The asymptotics of the approximation of differentiable functions by a Fourier series. Doklady Akad. Nauk Ross. 326 (1992), 31–34. English translation. :Russ. Acad. Sci., Dokl. Math. 46 (1993), 210–213.Google Scholar
[10]Müller, C.. Spherical harmonics. Lecture Notes in Mathematics, Vol. 17 (Springer-Verlag, 1966).CrossRefGoogle Scholar
[11]Stechkin, S. B.. An estimate of the remainder of the Taylor series for certain classes of analytic functions. Izv. Akad. Nauk SSSR Ser. Math. 17 (1953), 461–472.Google Scholar
[12]Stechkin, S. B.. Estimation of remainder of Fourier series for differentiable functions, Proc. Steklov Inst. Math. 145 (1980), 139–166.Google Scholar
[13]Szegö, G.. Orthogonal Polynomials, American Mathematical Society, New York, 1975.Google Scholar
[14]Tikhomirov, V. M.. A. N. Kolmogorov and approximation theory (Russian), Uspekhi Mat. Nauk 44 (1989), no. 1 (265), 83–122. English translation in Russian Math. Surveys 44 (1989), no. 1, 101–152.Google Scholar
[15]Trigub, R. M.. Multipliers of Fourier series and approximation of functions by polynomials in the spaces C and L (Russian). Dokl. Akad. Nauk SSSR 306 (1989), no. 2, 292–296. English translation: Soviet Math. Dokl. 39 (1989), no. 3, 494–498.Google Scholar
[16]Wang, K. Y. and Li, L. Q.. Harmonic Analysis and Approximation on the unit Sphere. (Science Press, 2000).Google Scholar