The Funk-Hecke Formula , Harmonic Polynomials , and Derivatives of Radial Distributions

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 R of the type Yk (


Introduction
Harmonic polynomials have a long and fruitful history in Mathematics and in Mathematical Physics, but it is noteworthy that there has been interest in several aspects of the relationship between harmonic polynomials and problems in distributional regularization and in the computation of distributional derivatives in recent years.Harmonic polynomials play a fundamental role in the ideas of the late professor Stora on convergent Feyman amplitudes, particularly in his work with Nikolov and Todorov [20]; this can be also seen in the recent article of Várilly and Gracia-Bondía [25].Parker [21] has pointed out the correct formulas obtained for multipole potentials built from harmonic polynomials, while the author has shown that such multipole potentials have remarkable properties with respect to 144 R. Estrada regularization and differentiation [6] and that several product formulas involving n dimensional delta functions simplify only for harmonic polynomials [7].
The aim of this article is to consider the computation of distributional derivatives of the type 1p ∇ (f (r)) , where f is a radial distribution in R n and where p is a polynomial in n variables of the special form p (x) = Y (x) |x| 2j , Y being a harmonic polynomial.Such derivatives, particularly in the case of regularizations of power potentials, are very important in Mathematical Physics [14,19,21], starting with the celebrated Frahm formulas [11] that have become standard material in textbooks [15].The general derivatives of any order of regularizations of power potentials are available [17], so that, in principle one could evaluate (1.1) for any polynomial p for this type of radial distributions f, but the formulas simplify substantially precisely if p is harmonic.
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.The usual Funk-Hecke formula has proved to be an indispensable aid in the study of multidimensional integral transforms, such as, for example, the Radon transform, from the pioneering work of Ludwig [18] to recent works [22].Our version of the formula is not only more general than the standard one, but can also be applied in other contexts, as we show in this article.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].

Notation
We shall write Notice that c 0,n = C = 2π n/2 /Γ (n/2) , is the surface area of the unit sphere S of R n .The space of homogeneous polynomials of degree k in n variables will be denoted as P k or P k (R n ) .The set of all polynomials in n variables will be denoted as P or as P (R n ) .In the space P (R n ) we consider the inductive limit topology [24,Chp. 14], so that it is an LF space.We denote by H k (R n ) the subspace of P k (R n ) formed by the harmonic homogeneous polynomials of degree k.We may also consider H k (S) , the set of restrictions to the unit sphere.The elements of H k (S) are usually called spherical harmonics, while those of H k (R n ) are referred to as solid harmonics.Notice that the restriction map H k (R n ) −→ H k (S) is in fact a bijection because 145 of the maximum principle for harmonic functions, and thus one may employ the simpler notation H k for this space2 .The space H = ∞ k=0 H k is the space of all harmonic polynomials, a closed subspace of the topological vector space P.
The dual space P ′ can be identified with the space of formal power series in n variables, and we endow it with the weak topology, which is exactly the topology of simple convergence of the coefficients [24].Similarly, H ′ can be identified with ∞ k=0 H k , with the product topology, or alternatively, with the space of formal series of the form ∞ k=0 Y k where Y k ∈ H k , with the topology of the simple convergence of each term of the series.When one thinks of the elements of H and of H ′ as objects defined on the sphere S, then it is many times true that spaces of functions and distributions, X, satisfy H ⊂ X ⊂ H ′ and H ⊂ X ′ ⊂ H ′ ; for example, H ⊂ L 2 (S) ⊂ H ′ , the elements of L 2 (S) being those series The projection from H ′ or any of its subspaces to H m will be denoted as The injection from H m to H or bigger spaces will be denoted as i m .In fact, where the kernel Z m (u, v) is the zonal harmonic of degree m, the reproducing kernel of the finite dimensional Hilbert space H k with structure as a subspace of L 2 (S) [2,9,22].Actually where the P m are appropiate multiples of the ultraspherical polynomials3 for dimension n [22, (A.6.13)].

The Funk-Hecke formula
The Funk-Hecke formula is a very useful tool in harmonic analysis 4 .The work of Funk and Hecke deals with the 3 dimensional case [10,13]; in its n dimensional form it was probably first given by Erdélyi [4] 5 .Here we shall follow the presentation of [22, Appendix A], and to some extent that of [3,Sect. 11.4].
The Funk-Hecke formula is usually written in the following way [22, Thm.A.34]: Notice that the constant λ k is the same for all spherical harmonics of the same degree k.Here we would like to give a general version of the formula that asks minimal regularity of the kernel f (u • v) , that is, by replacing the integral by a suitable evaluation f (u • v) , Y k (u) u we shall see that the Funk-Hecke formula continues to hold not only for distributional kernels, but actually for kernels that can be expressed as a formal power series.
Let us start by recalling the following result on invariant functions [22]: for some function of one variable G; actually if n ≥ 3 it is enough to ask invariance with respect to SO (n) .From this we obtain the following result on invariant transforms.
Lemma 3.1.Let G : H−→C (S) be a linear transform given by the formula if and only if for some G ∈ C (S) .If n ≥ 3 it is enough to ask invariance with respect to SO (n) .
Proof: If (3.4) holds, then (3.3) is obtained by a simple change of variables.Conversely, if (3.3) is satisfied, then for any τ ∈ O (n) the same change of variables gives for all f ∈ H, and the density of H in (C (S)) ✷ We shall improve this result to more general kernels g, but this weaker form will be useful in our analysis.
for any m, and thus the Lemma 3.1 gives that it comes from a continuous kernel g m (u, v) = G m (u • v) .If we now apply the Funk-Hecke formula for integrable kernels, we obtain that for each l, for all m, and this naturally gives ✷ We are now ready to give our version of the Funk-Hecke formula.
and for any where G is the formal series 7 the P k being the normalized ultraspherical polynomials (2.3) for dimension n.
Proof: Indeed, for any operator G : for some constants λ k , and this yields (3.7).The expansion (3.9) for the kernel G is obtained from (2.2). ✷ Notice that when the invariant operator G can be considered as acting from function or distribution spaces X and Y, with H ֒→ X ֒→ H ′ and H ֒→ Y ֒→ H ′ then the formal series G corresponds to a function or distribution, and the expansion (3.9) becomes convergent in a stronger sense.For instance, if G is an operator from D (S) to D ′ (S) then G converges distributionally.The operators that send given spaces X to Y can be characterized by studying the properties of the sequence We may now employ the Theorem 3.3 to obtain the form of several operators acting on polynomials in n variables.
converges in the topology of H ′ , since in H ′ the weak and strong topologies coincide [24]. 7Since the P k are polynomials, this is actually a formal power series.

R. Estrada
Proof: Apply the Funk-Hecke formula to the operator G defined on as required.✷ We can also rewrite these results in the ensuing way.
Proposition 3.5.Let T : P (R n ) −→P (R n ) be a linear operator that satisfies for a ∈ R \ {0} and for τ ∈ SO (n) .Then there are constants λ k,j such that Proof: It follows from the Proposition 3.4 since (3.11) implies that Notice that in a natural fashion one can consider P (R n ) as a subspace of then one may consider the family as a single operator T = {T n } ∞ n=1 , and one can write T {p} instead of T n {p} if p ∈ P (R n ) .In general the constants given in (3.12) will depend on n, that is, if However, the λ {n} k,0 are actually independent of n, and can be found rather easily.
Proposition 3.6.Let T = {T n } ∞ n=1 be a family of linear operators that sends and which is also invariant in the sense of (3.11).Then λ {n} k,0 = λ k,0 for all n, and whenever the indices i 1 , . . ., i k are all different.

Derivatives of Radial Distributions
149 with j = 0 holds for both λ {n} k,0 and λ {n ′ } k,0 , and hence we obtain ✷ We now present an example where all the computations are basically trivial, but -because of this -will allow the reader to appreciate the main ideas of our approach.
Example 3.1.Let p ∈ P k (R n ) and let f be a smooth function defined in (0, ∞) .Consider the Laplacian ∆ (f (r) p (x)) , r = |x| : it can be written, in several ways, as q (r, x) where q (ρ, x) is a polynomial in x whose coefficients are functions of ρ, but there is a unique expression of this form where for each ρ the polynomial q = q k belongs to P k (R n ) .Write The operator T n can be extended to P (R n ) by linearity.Notice, however, that the T n depend on n.Clearly T n is invariant in the sense of (3.11).Hence for some operators Λ {n} k,j that send smooth functions in (0, ∞) to smooth functions in (0, ∞) .Moreover, Λ {n} k,j (f (r)) = r −2j Λ {n} k,0 r 2j f (r) .On the other hand, the Λ {n} k,0 can be obtained by observing that T n {p}−∆ n (f (r)) p (x) is independent of n, so that we may take What happens if f is now a distribution of one variable, with support in [0, ∞)?It is not clear if the operators Λ {n} k,j can be defined for such distributions, since L (f ) cannot be defined as an element of D ′ (R) for all f ∈ D ′ (R) .However, there should be a way to extend the Λ {n} k,j to distributions since ∆ (f (r) p (x)) is a well defined distribution of D ′ (R n ) for any radial distribution f (r) of D ′ (R n ) ; we explain how this is achieved in Section 6.

Derivatives of smooth radial functions
In this section we shall apply the Propositions 3.5 and 3.6 to find the formulas for the computation of the action of certain differential operators on radial functions in R n .
We now assume that the radial functions are smooth, and then extend our analysis to distributional derivatives in the Section 6.Thus f will be a smooth R. Estrada function defined in some open subinterval of (0, ∞) , so that f (r) will be smooth in some annular region in R n .Theorem 4.1.Let L be the differential operator where

.3)
Proof: Let r be a fixed number in the domain of f.Consider the operator does not depend on n.Furthermore, T is invariant in the sense of (3.11).Hence (4.2) follows for some operators Therefore, by taking It is interesting to notice that the first expression for the operator Λ {n} k,j in (4.3) is, in a way, independent of n.Naturally, of course, Λ {n} k,j (f ) does depend on n if j > 0.
In order to appreciate the Theorem 4.1, it is instructive to consider a particular case, the derivatives of the power potentials f (r) = r λ .The distributional derivatives of any order of r λ were obtained by the author and are studied in the textbooks [17]; they generalize the important formulas of Frahm [11].Here we just consider the ordinary part of the formulas, that is, for r > 0; the delta part will be considered in Section 6.For example for derivatives of the forth order we have, Here and in similar formulas, ∇ N denotes the symmetric tensor of order N with components ∇ i1 • • • ∇ iN , x N is the tensor with components x i1 • • • x iN , while δ is the tensor of the second order with componets δ i1i2 .If S and T are symmetric tensors, then ST is their symmetric product, that is, the symmetrization of their tensor product S ⊗ T; the notation S Q will be used for the symmetric product of S with itself Q times.For example, x 2 δ is a forth order tensor with components {i,j}∪{k,l}={1,2,3,4} x i x j δ kl .
The Theorem 4.1 then yields In the case of order N, ∇ N r λ consists of a sum of [ [ N/2 ] ] + 1 terms, and so does, in general, p (∇) r λ if p ∈ P N (R n ) .However, if Y is a harmonic polynomial of degree N then Y (∇) r λ reduces to just the first term, Since [2,9] any polynomial p in R n can be expressed, uniquely, as a sum of terms of the form Then p is harmonic if and only if tr (A) = 0, so that we write (4.9)

Radial and related distributions
We shall now consider radial distributions and distributions that are radial multiples of a harmonic homogeneous polynomial.In order to fix the notation, we shall give the details in the spaces S (R n ) and S ′ (R n ) , but naturally the same considerations apply in the spaces D (R n ) and D ′ (R n ) , the spaces E (R n ) and E ′ (R n ) , or other dual pairs, without much change.A test function φ ∈ S (R n ) is called radial if it is a function of r, φ (x) = ϕ (r) , for some even function ϕ ∈ S (R) ; the space of all radial test functions of S (R n ) is denoted as S rad (R n ) .Similarly, we denote as S ′ rad (R n ) the space of all radial tempered distributions; a distribution , and this actually means that f (x) = f 1 (r) for some distribution of one variable f 1 .Notice, however, that while ϕ is uniquely determined by φ, for a given f ∈ S ′ rad (R n ) there are several possible distributions f 1 ∈ S ′ (R) .When n = 1 then S rad (R) and S ′ rad (R) become the spaces of even rapidly decreasing test functions and tempered distributions, respectively, and are also denoted as S even (R) and S ′ even (R) .Observe that the space S ′ rad (R n ) is naturally isomorphic to the dual space (S rad (R n )) ′ , that is to say, if the action of a radial distribution is known in all R. Estrada radial test functions, then it can be obtained for arbitrary test functions.Indeed, where φ ∈ S rad (R) is given as φ (x) = φ o (|x|) , φ o ∈ S even (R) being defined as Following [12], we shall denote by R n = r n−1 S even (R) .Also [5], if A is a subspace of S (R) we shall denote by A[0, ∞) the space of restrictions of elements of A to [0, ∞).The operator J : is an isomorphism [5].What this means is that when considering a radial distribution of S ′ (R n ) , we can express it, in a unique fashion, as f = F (r) , where It is interesting to observe that a radial distribution with support {0} should have the form N j=0 α j ∇ 2j δ (x) , for some N and some constants α j , 0 ≤ j ≤ N. Notice also the formulas where the c j,m are given in (2.1).Let now Y ∈ H k (R n ) be a solid harmonic and consider the multiplication map for some distribution of one variable F. As we explained, we can take F ∈ R ′ n [0, ∞), but actually we shall now show that a better choice will be F ∈ R ′ n+2k [0, ∞) since the operator M Y is not injective, but rather has a non trivial kernel.
Proof: Indeed, if f Y = 0 then supp f = {0} , so that, because f is radial, f (x) = N j=0 α j ∇ 2j δ (x) for some N and some constants α j .However

.6)
Derivatives of Radial Distributions

Derivatives of radial distributions
Our next task is to show that the Theorem 4.1 holds in the distributional sense for the derivatives of radial distributions.This will follow from a careful examination of the operator L. Proposition 6.1.The derivative operator d/dr, f → df /dr is a well defined operator from R ′ n [0, ∞) to R ′ n+1 [0, ∞).If q ∈ Z, then the operator M r q , f (r) → r q f (r) , is a well defined operator from R ′ n [0, ∞) to R ′ n−q [0, ∞).
Proof: The results follow immediately by considering the transpose operators.Indeed, we have d/dr T = −d/dr which sends R n+1 [0, ∞) to R n [0, ∞) and M r q T = M r q sends R n−q [0, ∞) to R n [0, ∞).✷ Therefore we obtain the following for the operator L.

R. Estrada
gives the most relevant formulas -several addition theorems for Bessel functions -citing the work of Bauer in 1859 for the 3 dimensional case and the work of Gegenbauer of 1874 in the general case.Interestingly, Erdélyi [4] actually employs these addition theorems in his study of the n dimensional Funk-Hecke formula.Such developments of the Fourier kernel are called Rayleigh expansions in the Physics literature, where they are still employed in the study of Fourier transforms [1].

Proposition 5 . 1 .
Let f be a radial distribution and let