Abstract
We give a version of the Funk-Hecke formula that holds with minimal assumptonsand apply it to obtain formulas for the distributional derivatives of radialdistributions in Rn of the typeYkrj(f (r)) ;where Yk is a harmonic homogeneous polynomial. We show that such derivatives havesimpler expressions than those of the form pr(f (r)) for a general polynomial p:
Highlights
Where f is a radial distribution in restriction map Hk (Rn) and where p is a polynomial in n variables of the special form p (x) = Y (x) |x|2j, Y being a harmonic polynomial
The main ingredients of our analysis are, first, a minimalistic version of the Funk-Hecke formula that holds for operators that transform polynomials into formal power series, the kernels themselves being formal power series, and which seems to have independent interest
Our second tool is a careful analysis of the spaces of distributions of the type f (r) Y (x), where f is radial and Y is a homogeneous harmonic polynomial, analysis that extends the study of radial distributions of [12] and [5]
Summary
Notice that c0,n = C = 2πn/2/Γ (n/2) , is the surface area of the unit sphere S of Rn. The space of homogeneous polynomials of degree k in n variables will be denoted as Pk or Pk (Rn). The set of all polynomials in n variables will be denoted as P or as P (Rn). We denote by Hk (Rn) the subspace of Pk (Rn) formed by the harmonic homogeneous polynomials of degree k. Hk is the space of all harmonic polynomials, a closed subspace of the topological vector space P. H′ as objects defined on the sphere S, it is many times true that spaces of functions and distributions, X, satisfy H ⊂ X ⊂ H′ and H ⊂ X′ ⊂ H′; for example, H ⊂ L2 (S) ⊂ H′, the elements of L2 (S) being those series. Where the Pm are appropiate multiples of the ultraspherical polynomials for dimension n [22, (A.6.13)]
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