# The Spherical Harmonic Spectrum of a Function with Algebraic Singularities

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## Abstract

The asymptotic behaviour of the spectral coefficients of a function provides a useful diagnostic of its smoothness. On a spherical surface, we consider the coefficients \(a_{l}^{m}\) of fully normalised spherical harmonics of a function that is smooth except either at a point or on a line of colatitude, at which it has an algebraic singularity taking the form *θ* ^{ p } or |*θ*−*θ* _{0}|^{ p } respectively, where *θ* is the co-latitude and *p*>−1. It is proven that each type of singularity has a signature on the rotationally invariant energy spectrum, \(E(l) = \sqrt{\sum_{m} (a_{l}^{m})^{2}}\) where *l* and *m* are the spherical harmonic degree and order, of *l* ^{−(p+3/2)} or *l* ^{−(p+1)} respectively. This result is extended to any collection of finitely many point or (possibly intersecting) line singularities of arbitrary orientation: in such a case, it is shown that the overall behaviour of *E*(*l*) is controlled by the gravest singularity. Several numerical examples are presented to illustrate the results. We discuss the generalisation of singularities on lines of colatitude to those on any closed curve on a spherical surface.

## Keywords

Spherical harmonics Singularity Spectrum Algebraic decay Darboux’s principle## Mathematics Subject Classification

33C55 65D15 42B05 65M70 78M22 41A25## Notes

### Acknowledgements

This work was supported by NERC grant NE/G014043/1 and benefitted from discussions with Rainer Hollerbach, Matt Daws, Evy Kersalé, Jitse Niesen, Andy Jackson, Frank Lowes and Avraham Sidi. Comments from two anonymous reviewers helped improve the manuscript.

## References

- 1.Abramowitz, M., Stegun, I.A.: Pocketbook of Mathematical Functions. Verlag Harri Deutsch, Frankfurt am Main (1984) MATHGoogle Scholar
- 2.Backus, G., Parker, R., Constable, C.: Foundations of Geomagnetism. CUP, Cambridge (1996) Google Scholar
- 3.Bain, M.: On the uniform convergence of generalized Fourier series. J. Inst. Math. Appl.
**21**, 379–386 (1978) MathSciNetMATHCrossRefGoogle Scholar - 4.Blakely, C., Gelb, A., Navarra, A.: An automated method for recovering piecewise smooth functions on spheres free from Gibbs oscillations. Sampl. Theory Signal Image Process.
**6**, 323–346 (2007) MathSciNetMATHGoogle Scholar - 5.Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2001) MATHGoogle Scholar
- 6.Boyd, J.P.: The Blasius function: computations before computers, the value of tricks, undergraduate projects, and open research problems. SIAM Rev.
**50**, 791–804 (2008) MathSciNetMATHCrossRefGoogle Scholar - 7.Boyd, J.P.: Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms. J. Eng. Math.
**63**, 355–399 (2009) MATHCrossRefGoogle Scholar - 8.Canuto, C., Quarteroni, A.: Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comput.
**38**, 67–86 (1982) MathSciNetMATHCrossRefGoogle Scholar - 9.Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006) MATHGoogle Scholar
- 10.Elliott, D.: The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function. Math. Comput.
**18**, 274–284 (1964) CrossRefGoogle Scholar - 11.Fox, L., Parker, I.B.: Chebyshev Polynomials in Numerical Analysis. Oxford University Press, London (1968) Google Scholar
- 12.Gage, K.S., Nastrom, G.D.: Theoretical interpretation of atmospheric wave number spectra of wind and temperature observed by commercial aircraft during GASP. J. Atmos. Sci.
**43**, 729–740 (1986) CrossRefGoogle Scholar - 13.Gelb, A.: The resolution of the Gibbs phenomenon for spherical harmonics. Math. Comput.
**66**, 699–717 (1997) MathSciNetMATHCrossRefGoogle Scholar - 14.Guo, B.-Y.: Spectral Methods and Their Applications. World Scientific Publishing, Singapore (1998) MATHCrossRefGoogle Scholar
- 15.Jackson, A.: Bounding the long wavelength crustal magnetic field. Phys. Earth Planet. Inter.
**98**, 283–302 (1996) CrossRefGoogle Scholar - 16.Jain, M.K., Chawla, M.M.: Estimation of the coefficients in the Legendre series expansion of a function. J. Math. Phys. Sci.
**1**, 247–260 (1967) MathSciNetMATHGoogle Scholar - 17.Kzaz, M.: Asymptotic expansion of Fourier coefficients associated to functions with low continuity. J. Comput. Appl. Math.
**114**, 217–230 (2000) MathSciNetMATHCrossRefGoogle Scholar - 18.Lighthill, M.J.: Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press, Cambridge (1958) MATHCrossRefGoogle Scholar
- 19.Livermore, P., Hollerbach, R.: Successive elimination of shear layers by a hierarchy of constraints in inviscid spherical-shell flows. J. Math. Phys. (2012). doi: 10.1063/1.4736990 Google Scholar
- 20.Mead, K.O., Delves, L.M.: On the convergence rate of generalized Fourier expansions. J. Inst. Math. Appl.
**12**, 247–259 (1973) MathSciNetMATHCrossRefGoogle Scholar - 21.Orszag, S.A.: Fourier-series on spheres. Mon. Weather Rev.
**102**, 56–75 (1974) CrossRefGoogle Scholar - 22.Sidi, A.: Asymptotic expansions of Legendre series coefficients for functions with interior and endpoint singularities. Math. Comput.
**80**, 1663–1684 (2011) MathSciNetMATHCrossRefGoogle Scholar - 23.Szegö, G.: Orthogonal polynomials, vol. 1, 4th edn. American Mathematical Society, Providence (1975) MATHGoogle Scholar
- 24.Viswanath, D., Şahutoğlu, S.: Complex singularities and the Lorenz attractor. SIAM Rev.
**52**, 478–495 (2010) CrossRefGoogle Scholar - 25.Weyl, H.: The Theory of Groups and Quantum Mechanics. Methuen & Co. LTD, London (1931) MATHGoogle Scholar