The resolution of the Gibbs phenomenon for spherical harmonics

Abstract

Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable oscillations over the entire sphere. Recently, methods for removing the Gibbs phenomenon for one-dimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is enough to recover an <italic>exponentially convergent</italic> approximation to the point values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in any subinterval in which the function is analytic. Here we take a similar approach, proving that knowledge of the first <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> spherical harmonic coefficients yield an <italic>exponentially convergent</italic> approximation to a spherical piecewise smooth function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis theta comma phi right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi> θ </mml:mi> <mml:mo>,</mml:mo> <mml:mi> ϕ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(\theta ,\phi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in any subinterval <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket theta 1 comma theta 2 right-bracket comma phi element-of left-bracket 0 comma 2 pi right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:msub> <mml:mi> θ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi> θ </mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mo>,</mml:mo> <mml:mi> ϕ </mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[\theta _1,\theta _2], \phi \in [0,2\pi ]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where the function is analytic. Thus we entirely overcome the Gibbs phenomenon.