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The Quarterly Journal of Mechanics and Applied Mathematics 1997 50(3):467-479; doi:10.1093/qjmam/50.3.467
© 1997 by Oxford University Press
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UNIFORM BOUNDS FOR Pnm(cos{Theta}) AND THE ABSOLUTE CONVERGENCE OF SERIES EXPANSIONS IN SPHERICAL SURFACE HARMONICS

R. DOUGLAS GREGORY

( Department of Mathematics, University of Manchester Oxford Road, Manchester M13 9PL )

An ‘arbitrary’ function f({Theta},{varphi}) (where r, {Theta}, {varphi} are a set of spherical polar co-ordinates) may be expanded in a series of spherical surface harmonics of the form


It is established how the rapidity of convergence of such an expansion series depends upon the smoothness of the function f({Theta}{varphi}) that is being expanded. This provides conditions under which the expansion series of f converges absolutely; asymptotic bounds on the expansion coefficients anm, bnm, are also deduced. Analytical properties are established for a function defined by the sum of a series of surface harmonics, whose coefficients are obtained by modifying the coefficients from the expansion of a given function. This is what is required in many applications. Simple uniform bounds for the functions Pnm(cos{Theta}) and their derivatives are also obtained and conditions are deduced under which a series of surface harmonics with generally assigned coefficients converges absolutely; analytical properties of the sum are also established.


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