Abstract
In this chapter we introduce spherical harmonics and study their properties. Most of the material of this chapter, except the last section, is classical. We strive for a succinct account of the theory of spherical harmonics. After a standard treatment of the space of spherical harmonics and orthogonal bases in the first section, the orthogonal projection operator and reproducing kernels, also known as zonal harmonics, are developed in greater detail in the second section, because of their central role in harmonic analysis and approximation theory. As an application of the addition formula, it is shown in the third section that there exist bases of spherical harmonics consisting of entirely zonal harmonics. The Laplace–Beltrami operator is discussed in the fourth section, where an elementary and self-contained approach is adopted. Spherical coordinates and an explicit orthonormal basis of spherical harmonics in these coordinates are presented the fifth section. These formulas in two and three variables are collected in the sixth section for easy reference, since they are most often used in applications.
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
Askey, R., Andrews, G., Roy, R.: Special functions. In: Encyclopedia of Mathematics and Its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
Atikinson, K., Han, W.: Spherical harmonics and approximation on the unit sphere: An introduction. In: Lecture Notes in Mathematics, vol. 2044. Springer, New York (2012)
Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory. Springer, New York (1992)
Dunkl, C.F., Xu, Y.: Orthogonal polynomials of several variables. In: Encyclopedia of Mathematics and Its Applications, vol. 81. Cambridge University Press, Cambridge (2001)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. McGraw-Hill, New York (1953)
Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press, New York (1996)
Helgason, S.: Groups and Geometric Analysis. Academic, New York (1984)
Hobson, E.W.: The Theory of Spherical and Elliptical Harmonics. Chelsea Publishing Co., New York (1955)
Müller, C.: Spherical harmonics. In: Lecture Notes in Mathematics, vol. 17. Springer, New York (1966)
Müller, C.: Analysis of Spherical Symmetries in Euclidean Spaces. Springer, New York (1997)
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)
Vilenkin, N.J.: Special functions and the theory of group representations. In: American Mathematical Society Translation of Mathematics Monographs, vol. 22. American Mathematical Society, Providence (1968)
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). Spherical Harmonics. 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_1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-6660-4_1
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)