The Funk-Hecke formula, harmonic polynomials, and derivatives of radial distributions

Abstract

We give a version of the Funk-Hecke formula that holds with minimal assumptons and apply it to obtain formulas for the distributional derivatives of radial distributions in Rn of the type Yk 􀀀 r j (f (r)) ; where Yk is a harmonic homogeneous polynomial. We show that such derivatives have simpler expressions than those of the form p 􀀀 r (f (r)) for a general polynomial p:

Keywords

• Harmonic polynomials
• distributional derivatives
• radial distributions