# 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.

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