On Pizzetti’s formula

Abstract

We give an extended Pizzetti asymptotic formula, namely, we show that if ϕ is smooth in an open set that contains the origin in $R^{n}$ and if p is a harmonic homogeneous polynomial of degree k, then $\begin{matrix} (0.1) & \int_{S} ϕ (ε ω) p (ω) d σ (ω) \sim C \sum_{m = 0}^{\infty} \frac{Δ^{m} p (\nabla) ϕ (0)}{W_{n, k, m}} ε^{k + 2 m}, \end{matrix}$ as $ε \to 0^{+}$ for some constants C and $W_{n, k, m}$ that are given in the text. We also show that the formula never holds, for all ϕ, if p is not harmonic.

We consider two procedures, one algebraic and the other analytic, to find the part of an object – a formal power series or a smooth function – that is a radial multiple of a given polynomial and show that the two constructions yield the same results for harmonic polynomials but not otherwise.

We also consider mean value type results for solutions of partial differential equations, including a version of Morera’s theorem that applies to locally integrable functions.

Keywords

Pizzetti’s formula harmonic polynomials mean value theorems

1. Introduction

In 1909 Pizzetti [15] studied the behavior of the integrals over spheres of small radius of a function ϕ that is smooth on a ball with center at the origin in $R^{3}$ , obtaining the asymptotic formula $\begin{matrix} (1.1) & \frac{1}{4 π} \int_{S} ϕ (ε ω) d σ (ω) \sim \sum_{m = 0}^{\infty} \frac{Δ^{m} ϕ (0)}{(2 m + 1)!} ε^{2 m}, ε \to 0^{+} . \end{matrix}$ The corresponding formula in $R^{n}$ takes the form [16] $\begin{matrix} (1.2) & \frac{1}{C} \int_{S} ϕ (ε ω) d σ (ω) \sim \sum_{m = 0}^{\infty} \frac{Δ^{m} ϕ (0)}{2^{m} m! n (n + 2) \dots (n + 2 m - 2)} ε^{2 m}, \end{matrix}$ as $ε \to 0^{+}$ , where we use the notation $S$ for the unit sphere in $R^{n}$ and $\begin{matrix} (1.3) & C = \frac{2 π^{n / 2}}{Γ (n / 2)} \end{matrix}$ for its area.

Pizzetti’s asymptotic formula has been employed in several areas, particularly in mean value theorems for partial differential equations [6,12,17,23] and in the study of products and asymptotic behavior of generalized functions [2,3,13]. It has been generalized in several directions, including [1] where integrals over other surfaces are considered, [21] in the framework of manifolds, and [22] where the function ϕ can be a thick test function.

The aim of this article is to consider an interesting generalization, namely we show that when p is a harmonic homogeneous polynomial of degree k in n variables, then $\begin{matrix} (1.4) & \frac{1}{C} \int_{S} p (ω) ϕ (ε ω) d σ (ω) \sim \sum_{m = 0}^{\infty} \frac{Δ^{m} p (\nabla) ϕ (0)}{W_{n, k, m}} ε^{k + 2 m}, \end{matrix}$ where $W_{n, 0, 0} = 1$ and $W_{n, k, m} = 2^{m} m! n (n + 2) \dots (n + 2 k + 2 m - 2)$ if $k + m > 0$ , that is, $\begin{matrix} (1.5) & W_{n . k, m} = \frac{2^{2 m + k} m! Γ (k + m + n / 2)}{Γ (n / 2)} . \end{matrix}$ The original Pizzetti expansion corresponds to the case when $p = 1$ . It is not hard to obtain the expansion of the integrals $\int_{S} p (ω) ϕ (ε ω) d σ (ω)$ as $ε \to 0^{+}$ for a general polynomial p, homogeneous of degree k, but the formula (1.4) will not be valid unless the polynomial is harmonic. In fact, if $f \in D^{'} (S)$ is a distribution on the sphere, we have [9] $\begin{matrix} (1.6) & {⟨ f (ω), ϕ (ε ω) ⟩}_{D^{'} (S) \times D (S)} \sim \sum_{j = 0}^{\infty} (\sum_{α \in N^{n}, | α | = j} \frac{μ_{α} \nabla^{α} ϕ (0)}{α!}) ε^{j} \end{matrix}$ for any $ϕ \in D (B)$ , B being a ball with center at the origin, where $μ_{α} = ⟨ f (ω), ω^{α} ⟩$ are the moments of the distribution $f (ω) δ (r - 1)$ of $D^{'} (R^{n})$ . No further simplification seems possible for a general f, not even for a general polynomial, so that the fact that (1.4) could be obtained is rather surprising. Additionally we observe that while in general the expansion of ${⟨ f (ω), ϕ (ε ω) ⟩}_{D^{'} (S) \times D (S)}$ contains both even and odd powers and starts with the power $j = 0$ , the special case of a harmonic homogeneous polynomial of degree k has only powers of the form k, $k + 2$ , $k + 4$ , $\dots$ .

The plan of the article is the following. We start by clarifying our notation in Section 2. Then in the Section 3 we construct the radial part of a formal power series and explain what a radial free formal power series is, by using an algebraic construction and then we compare it to the analytic construction, showing that Pizzetti’s formula provides the bridge between them. Next, we construct the part of a series that is a radial multiple of a given polynomial algebraically in Section 4 and analytically in Section 5, obtaining the curious fact that the two constructions yield the same results if and only if the polynomial is a harmonic polynomial. We also give an algebraic decomposition of formal power series that turns out to be equivalent to the well known Fourier–Laplace series. The extended Pizzetti formula is given in the Theorem 5.1 and in distributional form in the Proposition 6.1, where an alternative proof is presented. In the last section we apply our ideas to obtain mean value results for solutions of partial differential equations.

2. Notation

In this article we employ the word smooth to mean $C^{\infty}$ .

We denote as $P_{k}$ the space of homogeneous polynomials of degree k in n variables. There is an inner product in $P_{k}$ defined in terms of the coefficients as $\begin{matrix} (2.1) & {p, q} = \sum_{| α | = k} α! a_{α} {\overline{b}}_{α}, \end{matrix}$ if $p (x) = \sum_{| α | = k} a_{α} x^{α}$ and $q (x) = \sum_{| α | = k} b_{α} x^{α}$ . Notice that ${p, q}$ actually equals the following constant function, ${p, q} = p (\nabla) \overline{q (x)}$ , where $\nabla = {(\partial / \partial x_{i})}_{i = 1}^{n}$ is the gradient. We denote as $P = ⨁_{k = 0}^{\infty} P_{k}$ the space of all polynomials in n variables; each $P_{k}$ is a Hilbert space, while $P$ is a pre-Hilbert space with the inner product ${,}$ , but usually one endows $P$ with the inductive limit topology [19].

We denote as $H_{k}$ the subspace $P_{k}$ formed by the harmonic homogeneous polynomials of degree k.

The space of formal power series in n variables will be denoted as $C [[x_{1}, \dots, x_{n}]]$ . Usually one considers it as a topological vector space by endowing it with the topology of simple convergence of the coefficients of the series, and with this topology the spaces $C [[x_{1}, \dots, x_{n}]]$ and $P$ , with the inductive limit topology, are dual spaces [19]; the duality can be given by the formula $\begin{matrix} (2.2) & ⟨ S, q ⟩ = \sum_{α \in N^{n}} α! a_{α} b_{α}, \end{matrix}$ if $S = \sum_{α \in N^{n}} a_{α} x^{α} \in C [[x_{1}, \dots, x_{n}]]$ and $q = \sum_{α \in N^{n}} b_{α} x^{α} \in P$ , (2.2) being a finite sum. Since $P \subset C [[x_{1}, \dots, x_{n}]]$ we can actually consider the evaluation $⟨ p, q ⟩$ if both p and q are polynomials, and clearly, $\begin{matrix} (2.3) & ⟨ p, q ⟩ = {p, \overline{q}}, \end{matrix}$ in that case.

3. Radial and radial-free objects

It is very clear what a radial polynomial is, namely a polynomial in $r^{2}$ . Thus a homogeneous polynomial of degree k, $p \in P_{k}$ , is radial if k is even, $k = 2 m$ , and $p = c r^{k}$ for some constant c. A much more interesting question is which polynomials can be considered to be “radial-free.” Naturally any $p \in P_{k}$ is radial-free if k is odd, but what if k is even?

In order to understand the situation we may start when $k = 2$ , and try to find the radial part $ρ (p) = c_{p} r^{2}$ of $p \in P_{2}$ . Symmetry considerations tell us that $ρ (x_{1}^{2}) = \dots = ρ (x_{n}^{2})$ and that $ρ (x_{i} x_{j}) = ρ (x_{1} x_{2})$ if $i \neq j$ . Also applying a rotation of angle $π / 4$ we see that $ρ ({(x_{1} + x_{2})}^{2}) = 2 ρ (x_{1}^{2})$ . Hence $ρ (x_{1} x_{2}) = 1 / 2 [ρ ({(x_{1} + x_{2})}^{2}) - ρ (x_{1}^{2}) - ρ (x_{2}^{2})] = 0$ . Therefore since $ρ (r^{2}) = r^{2}$ , we obtain $ρ (x_{1}^{2}) = \dots = ρ (x_{n}^{2}) = r^{2} / n$ and thus $\begin{matrix} (3.1) & ρ (\sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i j} x_{i} x_{j}) = (\frac{1}{n} \sum_{i = 1}^{n} a_{i i}) r^{2} . \end{matrix}$

We can rewrite this formula by employing the inner product (2.1) as $\begin{matrix} (3.2) & ρ (p) = \frac{{p, r^{2}}}{{r^{2}, r^{2}}} r^{2} . \end{matrix}$ This we recognize as the formula for the projection in the direction of $r^{2}$ in the inner product space $P_{2}$ . We therefore copy this in $P_{2 m}$ , namely, we put $\begin{matrix} (3.3) & ρ (p_{2 m}) = \frac{{p_{2 m}, r^{2 m}}}{{r^{2 m}, r^{2 m}}} r^{2 m}, p_{2 m} \in P_{2 m}, \end{matrix}$ and if $p \in P$ , $p = \sum_{k = 0}^{N} p_{k}$ , $p_{k} \in P_{k}$ , we put $ρ (p) = \sum_{m = 0}^{[[N / 2]]} ρ (p_{2 m})$ . Defined in this way, $ρ (p)$ is the radial polynomial that best approximates p in the norm induced by the inner product ${,}$ , $\begin{matrix} (3.4) & {p - ρ (p), p - ρ (p)} ⩽ {p - q, p - q} for q radial . \end{matrix}$

Now we can answer the question of when $p \in P$ is radial free. Indeed, this will be the case if p is orthogonal to the set of radial polynomials, that is, if $\begin{matrix} (3.5) & {p, r^{2 m}} = 0, m = 0, 1, 2, \dots . \end{matrix}$ But we have the formula [10] $\begin{matrix} (3.6) & {p, x_{i} q} = {\nabla_{i} p, q}, \end{matrix}$ that yields $\begin{matrix} (3.7) & {p, r^{2 m}} = {Δ^{m} p, 1} = Δ^{m} p |_{x = 0}, \end{matrix}$ where $Δ = \nabla^{2}$ is the Laplacian. Consequently we obtain the following characterization of radial free polynomials.

Proposition 3.1.
A polynomial $p \in P$ is radial free if and only if $\begin{matrix} (3.8) & Δ^{m} p |_{x = 0} = 0, m = 0, 1, 2, \dots . \end{matrix}$

We can also rewrite the expressions for the radial part of a polynomial if we employ (3.7). Indeed, $\begin{matrix} (3.9) & {r^{2 m}, r^{2 m}} = Δ^{m} r^{2 m} = 2^{m} m! n (n + 2) \dots (n + m - 2) = W_{n, 0, m}, \end{matrix}$ and this yields the ensuing expression for the radial part of a polynomial.
Proposition 3.2.
If $p \in P$ then its radial part is $\begin{matrix} (3.10) & ρ (p) = \sum_{m = 0}^{\infty} \frac{Δ^{m} p |_{x = 0}}{2^{m} m! n (n + 2) \dots (n + m - 2)} r^{2 m} . \end{matrix}$

It should be clear, in fact, that formula (3.10) for the radial part can be applied if p is replaced by a formal power series $S \in C [[x_{1}, \dots, x_{n}]]$ : $\begin{matrix} (3.11) & ρ (S) = \sum_{m = 0}^{\infty} \frac{⟨ S, r^{2 m} ⟩}{W_{n, 0, m}} r^{2 m}, \end{matrix}$ and that actually $⟨ S, r^{2 m} ⟩$ can be understood as $Δ^{m} S |_{x = 0}$ .

If, on the other hand, φ is a smooth function in $R^{n}$ , then it would be radial if it is of the form $φ (x) = φ_{0} (r)$ , for some function of one variable $φ_{0}$ . How do we get the radial part of a general smooth function ϕ defined in $R^{n}$ ? Well, we just employ the natural definition [11], namely, $\begin{matrix} (3.12) & \tilde{ρ} (ϕ) (r) = \frac{1}{C} \int_{S} ϕ (r ω) d ω . \end{matrix}$ We then say that ϕ is “radial-free” if $\tilde{ρ} (ϕ) = 0$ for all r.

If we start with a smooth function ϕ defined in $R^{n}$ we can find its Taylor series at the origin, a formal power series $T (ϕ) \in C [[x_{1}, \dots, x_{n}]]$ and then find its radial part, $ρ (T (ϕ))$ , employing (3.11); or we can first find its radial part $\tilde{ρ} (ϕ)$ and then find the Taylor series at the origin of this function of one variable, $T (\tilde{ρ} (ϕ))$ . Will we get the same result? Pizzetti’s formula says that this is exactly the case!, since $\begin{matrix} (3.13) & \frac{1}{C} \int_{S} ϕ (ε ω) d ω \sim \sum_{m = 0}^{\infty} \frac{Δ^{m} ϕ |_{x = 0}}{2^{m} m! n (n + 2) \dots (n + m - 2)} ε^{2 m}, \end{matrix}$ as $ε \to 0^{+}$ , means precisely that $\begin{matrix} (3.14) & ρ (T (ϕ)) = T (\tilde{ρ} (ϕ)) . \end{matrix}$
4. Decomposition of a formal power series

Let $S \in C [[x_{1}, \dots, x_{n}]]$ be a formal power series in n variables. We can of course write it as $\begin{matrix} (4.1) & S = \sum_{α \in N^{n}} a_{α} x^{α}, \end{matrix}$ but we would like to rewrite this series in a different way. Indeed, we can first find the radial part $ρ (S)$ and consider $S - ρ (S)$ , a power series without a constant term.

Then we can find the part of the series $S_{1} = S - ρ (S)$ that is a multiple of $x_{1}$ , say $x_{1} T$ , find the radial part of T, say $ρ_{x_{1}} (S) = ρ (T)$ and substract this from $S - ρ (S)$ to obtain $S - ρ (S) - x_{1} ρ_{x_{1}} (S)$ , then do the same with $x_{2}$ , then $x_{3}$ and so on, obtaining a series of the form $S_{2} = S - ρ (S) - \sum_{i = 1}^{n} ρ_{x_{i}} (S) x_{i}$ , which is now a series without terms of orders 0 or 1.

Then we take an ordered basis of the polynomials of degree 2 that have not been removed, say $B_{2} = {p_{1}^{(2)}, \dots, p_{d_{2}}^{(2)}}$ , remove the parts of $S_{2}$ that are radial multiples of the elements of the basis, in order, to obtain a new series $S_{3} = S - ρ (S) - \sum_{i = 1}^{n} ρ_{x_{i}} (S) x_{i} - \sum_{p_{j}^{(2)} \in B_{2}} ρ_{p_{j}^{(2)}} (S) p_{j}^{(2)}$ that does not have terms of orders smaller than 3. And we repeat this process again and again to obtain the expression $\begin{matrix} (4.2) & S = R_{0} + R_{1} + R_{2} + R_{3} + \dots, \end{matrix}$ where $R_{k}$ is the part of S formed by radial multiples of polynomials of degree k that have not been removed in the previous steps.

Naturally several questions arise. Indeed, do the series $R_{k}$ depend on only S, or do they depend on the previously constructed series $R_{0}, \dots, R_{k - 1}$ ? Do they depend on the basis’ $B_{1} = {x_{1}, \dots, x_{n}}$ , $B_{2}$ , $B_{3}$ , $\dots$ or their order? And even a more basic question, namely, what is the part of the series S that is a radial multiple of a given homogeneous polynomial p, like the terms $ρ_{p_{j}^{(2)}} (S) p_{j}^{(2)}$ , for instance? Therefore we first explain the meaning of this.

4.1. The part of a series that is a radial multiple of a given polynomial

Notice that it is clear what one means by the part of the series S that is a radial multiple of $x_{i}$ , or of $x_{i} x_{j}$ , and so on, but what is the part of the series S that is a radial multiple of a polynomial such as $x_{1} + x_{2}$ or such as $x_{1}^{2} - 2 x_{2}^{2} ?$ We can give the answer in two equivalent ways. Fix $p \in P_{k}$ and let $q \in P$ . Then we can write $\begin{matrix} (4.3) & q = p u + v \end{matrix}$ for polynomials $u \in P$ and $v \in P$ in infinitely many ways, but there is a unique decomposition1

¹
The nice proof of this fact is well known. See [18, Theorem A.9] or [10, Proposition (2.47)], for instance.

if we ask that v satisfies the extra condition

\begin{matrix} (4.4) & \overline{p} (\nabla) v = 0 . \end{matrix}

This decomposition is what we do when we take the part of a polynomial that is multiplied by

x_{i}

, or by

x_{i} x_{j}

, etc., and thus we can do it in the same way for any p. Once q is decomposed like this, we take the radial part of u, to obtain that the part of q that is a radial multiple of p is then

\begin{matrix} (4.5) & ρ_{p} (q) p = ρ (u) p . \end{matrix}

Alternatively, we can define

ρ_{p} (q)

by asking

ρ_{p} (q) p

to be the projection of q onto the subspace of

P

consisting of radial multiples of p – with the inner product

{,}

– so that if

q_{k + 2 m} \in P_{k + 2 m}

\begin{matrix} (4.6) & ρ_{p} (q_{k + 2 m}) = \frac{{q_{k + 2 m}, r^{2 m} p}}{{r^{2 m} p, r^{2 m} p}} r^{2 m} = \frac{Δ^{m} \overline{p} (\nabla) q_{k + 2 m}}{{r^{2 m} p, r^{2 m} p}} r^{2 m} . \end{matrix}

Notice that this yields that a general

q \in P

is free from radial multiples of p if and only if

\begin{matrix} (4.7) & Δ^{m} \overline{p} (\nabla) q |_{x = 0} = 0, m = 0, 1, 2, \dots . \end{matrix}

Furthermore, we obtain the formula

\begin{matrix} (4.8) & ρ_{p} (q) = \sum_{m = 0}^{\infty} \frac{Δ^{m} \overline{p} (\nabla) q |_{x = 0}}{{r^{2 m} p, r^{2 m} p}} r^{2 m}, \end{matrix}

that can actually be applied even if q is replaced by a formal power series.

4.2. Back to the construction

Notice that in the particular case when both p and q belong to $P_{k}$ then q is free of radial multiples of p precisely when they are orthogonal, ${q, p} = \overline{p} (\nabla) q = 0$ . This allow us to answer the question about the influence of the order of the elements of the basis $B_{k}$ in our construction – assuming that $R_{0}, \dots, R_{k - 1}$ are known – indeed, the terms $ρ_{p_{j}^{(k)}} (S) p_{j}^{(k)}$ will be well defined only when $B_{k}$ is an orthogonal basis of $(P_{k}, {,})$ , since otherwise they depend on the order of $B_{k}$ , and a notation like $ρ_{p_{j}^{(k)}, B_{k}} (S) p_{j}^{(k)}$ would actually be needed.

Hence, as suspected, $ρ_{x_{i}, B_{1}} (S) x_{i}$ can be computed independently of the order in $B_{1} = {x_{1}, \dots, x_{n}}$ . Interestingly, even if we consider another basis $B_{1}^{'}$ of $P_{1}$ the corresponding sum $\sum_{p_{j}^{(1)} \in B_{1}^{'}} ρ_{p_{j}^{(1)}, B_{1}^{'}} (S) p_{j}^{(1)}$ also equals $\sum_{i = 1}^{n} ρ_{x_{i}} (S) x_{i}$ . This means that $R_{1}$ is well defined. Actually an inductive process allows us to see that all $R_{k}$ are well defined functions of S, that is, they do not depend on the previously constructed series $R_{0}, \dots, R_{k - 1}$ , since even if the individual terms $ρ_{p_{j}^{(k)}, B_{k}} (S) p_{j}^{(k)}$ depend on the order of $B_{k}$ , the sum $R_{k} = \sum_{p_{j}^{(k)} \in B_{k}} ρ_{p_{j}^{(k)}, B_{k}} (S) p_{j}^{(k)}$ is in fact independent of the basis $B_{k}$ .

Our analysis has another interesting consequence, namely, the subspace of $P_{k}$ consisting of the homogeneous polynomials of degree k not removed by $R_{0}, \dots, R_{k - 1}$ , can be given a direct description, not only as the set of polynomials that are free of radial multiples of polynomials of lower degree.

Proposition 4.1.
A polynomial $p \in P_{k}$ is free of radial multiples of polynomials of lower degree if and only if it is harmonic, $p \in H_{k}$ .
Proof.
The result is clear for $k = 0$ and $k = 1$ . The case $k = 2$ already suggests how the reasoning should go, since the Proposition 3.1 tell us that $p \in P_{2}$ is radial free if and only if ${p, r^{2}} = Δ p = 0$ . Let $p \in P_{k}$ ; while it is clear that p is free of radial multiples of polynomials of $P_{j}$ if $k + j$ is odd, it will be free from multiples of $q \in P_{k - 2 m}$ if and only if ${r^{2 m} q, p} = 0$ . But if $q \in H_{k - 2 m}$ then [7, Proposition 3.3] $\begin{matrix} (4.9) & {r^{2 m} q, p} = W_{n, k - 2 m, m} (q, p), \end{matrix}$ where the inner product $(q, p)$ is $\begin{matrix} (4.10) & (q, p) = \frac{1}{C} \int_{S} q (ω) \overline{p} (ω) d σ (ω) . \end{matrix}$ Hence if q is harmonic, ${r^{2 m} q, p} = 0$ if and only if $(q, p) = 0$ .

We thus obtain that p is free of radial multiples of harmonic polynomials of lower degree if and only if p is orthogonal to all harmonic polynomials of lower degree with respect to the inner product (4.10), and this property characterizes harmonic polynomials [5]. Our result follows since for any j the polynomials of the form $q r^{2 m}$ for $q \in H_{j - 2 m}$ and $0 ⩽ 2 m ⩽ j$ span $P_{j}$ , so that being free of radial multiples of polynomials of lower degree is the same as being free of radial multiples of harmonic polynomials of lower degree. □

4.3. Summary

We now summarize the main points in the decomposition of formal power series. If $S \in C [[x_{1}, \dots, x_{n}]]$ we can construct formal power series $R_{0}, R_{1}, R_{2}, \dots$ such that $\begin{matrix} (4.11) & S = \sum_{k = 0}^{\infty} R_{k}, \end{matrix}$ as follows: $R_{k}$ , a series that starts with degree k, is the part of S consisting of the radial multiples of homogeneous polynomials of degree k, but free of radial multiples of homogeneous polynomials of degree less than k. The series $R_{k}$ can be constructed without needing to know $R_{0}, \dots, R_{k - 1}$ ; in fact, if $B_{k}$ is an orthogonal basis for $H_{k}$ , the space of harmonic homogeneous polynomials of degree k, – with either the inner product ${,}$ or $(,)$ , it is the same because of (4.9) – then $\begin{matrix} (4.12) & R_{k} = \sum_{p_{j}^{(k)} \in B_{k}} ρ_{p_{j}^{(k)}} (S) p_{j}^{(k)}, \end{matrix}$ where the part of S that is a radial multiple of a harmonic homogeneous polynomial $p \in H_{k}$ is $ρ_{p} (S) p$ , and2

²
If $p \in P_{k}$ then $ρ_{p} (S)$ is given by (4.8), but not (4.13), that applies only if p is harmonic.

\begin{matrix} (4.13) & ρ_{p} (S) = \frac{1}{(p, p)} \sum_{m = 0}^{\infty} \frac{Δ^{m} \overline{p} (\nabla) S |_{x = 0}}{W_{n, k, m}} r^{2 m} = \frac{1}{{p, p}} \sum_{m = 0}^{\infty} \frac{Δ^{m} \overline{p} (\nabla) S |_{x = 0}}{W_{n + 2 k, 0, m}} r^{2 m} . \end{matrix}

5. Analytic constructions

We shall now consider the analytic constructions that correspond to our decomposition of formal power series presented in the previous section. As we shall see, not only are those constructions possible, but actually the objects obtained are very well known. In the process we study the relationship between the algebraic and analytic decompositions, obtaining an extended Pizzetti formula.

Let us start with radial multiples of a given function. Let $φ (x)$ be a given smooth function defined for $a < | x | < b$ . If ϕ is another smooth function with the same domain, then the part of ϕ that is a radial multiple of φ is ${\tilde{ρ}}_{φ} (ϕ) (| x |) φ (x)$ , where $\begin{matrix} (5.1) & {\tilde{ρ}}_{φ} (ϕ) (r) = \frac{1}{\int_{S} | φ (r ω) |^{2} d σ (ω)} \int_{S} ϕ (r ω) \overline{φ} (r ω) d σ (ω) . \end{matrix}$ The function ${\tilde{ρ}}_{φ} (ϕ) (| x |) φ (x)$ is the projection of ϕ into the space of radial multiples of φ in any of the pre-Hilbert spaces of smooth functions in $a < r < b$ with inner product $\begin{matrix} (5.2) & {(f, g)}_{c, d} = \int_{c}^{d} \int_{S} f (r ω) \overline{g} (r ω) d σ (ω) d r, \end{matrix}$ if $a < c < d < b$ .

Suppose now that $p \in P_{k}$ and ϕ is smooth for $| x | < r_{0}$ . Then we can find the radial part ${\tilde{ρ}}_{p} (ϕ) (r)$ and try to find its asymptotic behavior as $r \to 0^{+}$ but we can also find the radial part $ρ_{p} (T)$ of the Taylor series T of ϕ at the origin. Will we get the same? We now show that when p is harmonic and only then, the results are the same. This is our Extended Pizzetti formula.

Theorem 5.1.
Let ϕ be smooth for $| x | < r_{0}$ . If $p \in H_{k}$ then $\begin{matrix} (5.3) & \frac{1}{C} \int_{S} ϕ (ε ω) p (ω) d σ (ω) \sim \sum_{m = 0}^{\infty} \frac{Δ^{m} p (\nabla) ϕ (0)}{W_{n, k, m}} ε^{k + 2 m}, \end{matrix}$ as $ε \to 0^{+}$ . The formula never holds, for all ϕ, if $p \in P_{k} ∖ H_{k}$ .
Proof.
Suppose $p \in H_{k}$ . Let us write $ϕ (x) = \sum_{j = 0}^{N} q_{j} (x) + O (| x |^{N + 1})$ , where $q_{j} \in P_{j}$ . Then $\begin{matrix} (5.4) & \int_{S} ϕ (ε ω) p (ω) d σ (ω) = \sum_{j = 0}^{N} (\int_{S} q_{j} (ω) p (ω) d σ (ω)) ε^{j} + O (ε^{N + 1}), \end{matrix}$ and since $\int_{S} q_{j} (ω) p (ω) d σ (ω) = 0$ unless $j = k + 2 m$ for some m, $\begin{matrix} (5.5) & \begin{matrix} \frac{1}{C} \int_{S} ϕ (ε ω) p (ω) d σ (ω) & = \sum_{m = 0}^{M} (q_{k + 2 m}, \overline{p}) ε^{k + 2 m} + O (ε^{k + 2 M + 1}) \\ = \sum_{m = 0}^{M} \frac{{q_{k + 2 m}, r^{2 m} \overline{p}}}{W_{n, k, m}} ε^{k + 2 m} + O (ε^{k + 2 M + 1}) \\ = \sum_{m = 0}^{M} \frac{Δ^{m} p (\nabla) q_{k + 2 m}}{W_{n, k, m}} ε^{k + 2 m} + O (ε^{k + 2 M + 1}) \\ = \sum_{m = 0}^{M} \frac{Δ^{m} p (\nabla) ϕ (0)}{W_{n, k, m}} ε^{k + 2 m} + O (ε^{k + 2 M + 1}), \end{matrix} \end{matrix}$ if we take (4.9) into account. The extended Pizzetti formula has been established.

When $p \in P_{k} ∖ H_{k}$ then $\int_{S} q_{j} (ω) p (ω) d σ (ω) \neq 0$ for some $j < k$ , and some $q_{j} \in P_{j}$ so that the development of $\int_{S} ϕ (ε ω) p (ω) d σ (ω)$ as $ε \to 0^{+}$ is not of the form $O (ε^{k})$ whenever this $q_{j}$ appears in the expansion of ϕ, and consequently (5.3) cannot hold. □

We can also express the extended Pizzetti formula in the following way.
Proposition 5.2.
Let ϕ be smooth for $| x | < r_{0}$ . If $p \in H_{k}$ then ${\tilde{ρ}}_{p} (ϕ) (r)$ is a smooth function of r for $| r | < r_{0}$ , whose Taylor series at the origin is $ρ_{p} (T)$ , where T is the Taylor series of ϕ at the origin. If $p \in P_{k} ∖ H_{k}$ then there are functions ϕ, smooth for $| x | < r_{0}$ , for which ${\tilde{ρ}}_{p} (ϕ) (r)$ is not smooth at $r = 0$ .

Next we consider the analytic analog of the decomposition (4.11) of a formal power series. Indeed, we choose the same family of orthogonal basis $B_{k}$ of $H_{k}$ . Then we decompose a given smooth function ϕ as a sum $\begin{matrix} (5.6) & χ_{0} + χ_{1} + χ_{2} + \dots, \end{matrix}$ where $\begin{matrix} (5.7) & χ_{k} = \sum_{p_{j}^{(k)} \in B_{k}} {\tilde{ρ}}_{p_{j}^{(k)}} (ϕ) p_{j}^{(k)}, \end{matrix}$ and where ${\tilde{ρ}}_{p_{j}^{(k)}} (ϕ)$ is given in (5.1). The Proposition 5.2 tell us that each $χ_{k}$ is smooth at the origin and its Taylor series is exactly $R_{k}$ .

The reader will probably recognize the construction (5.6)–(5.7)! It is the well known Fourier–Laplace series for ϕ, of importance in several areas such as integral geometry (see [18], for instance).
6. Distributional expansions

The study of the asymptotic behavior of distributions has received a lot of attention in recent years [9,14,20]. Asymptotic expansions of integrals and series are many times very conveniently expressed in terms of the asymptotic expansion of distributions; for instance one can find a distributional version of Pizzetti’s formula in [9, Example 94], namely $\begin{matrix} (6.1) & δ (S_{ε}) \sim \sum_{m = 0}^{\infty} \frac{c_{m, n} \nabla^{2 m} δ (x)}{(2 m)!} ε^{2 m - n + 1}, \end{matrix}$ as $ε \to 0^{+}$ , where $c_{m, n} = 2 Γ (m + 1 / 2) π^{(n - 1) / 2} / Γ (m + n / 2)$ and where $δ (S_{ε})$ is the distribution of uniform mass concentrated on a sphere of radius ε, that is, $δ (S_{ε}) = δ (r - ε)$ . Naturally (6.1) and (1.2) are equivalent, since $c_{m, n} / (2 m)! = C / W_{n, 0, m}$ .

We can also give the distributional version of our extended Pizzetti formula.

Proposition 6.1.
Let $p \in H_{k}$ . Then $\begin{matrix} (6.2) & p (ω) δ (S_{ε}) = p (ω) δ (r - ε) \sim {(- 1)}^{k} C \sum_{m = 0}^{\infty} \frac{Δ^{m} p (\nabla) δ (x)}{W_{n, k, m}} ε^{k + 2 m + n - 1}, \end{matrix}$ as $ε \to 0^{+}$ in the space $D^{'} (B)$ , for any ball B centered at the origin.
Proof.
Indeed, the result follows immediately from the Theorem 5.1 since evaluation of (6.2) at a test function $ϕ \in D (B)$ is, after obvious manipulations, the extended Pizzetti formula (5.3). We can also give an alternative proof – that thus provides another proof of the Theorem 5.1 – as follows. Starting from (6.1) we obtain $\begin{matrix} (6.3) & ε^{k} p (ω) δ (S_{ε}) = p (x) δ (S_{ε}) \sim \sum_{m = 0}^{\infty} \frac{c_{m, n} p (x) \nabla^{2 m} δ (x)}{(2 m)!} ε^{2 m - n + 1} . \end{matrix}$ But if $p \in H_{k}$ then [8] $p (x) \nabla^{2 m} δ (x) = 0$ if $k > m$ , while $\begin{matrix} (6.4) & p (x) \nabla^{2 m} δ (x) = \frac{{(- 1)}^{k} 2^{k} m!}{(m - k)!} p (\nabla) \nabla^{2 m - 2 k} δ (x), \end{matrix}$ if $k ⩽ m$ . Substitution of (6.4) in (6.3) immediately gives $\begin{array}{l} ε^{k} p (ω) δ (S_{ε}) & \sim {(- 1)}^{k} \sum_{m^{'} = k}^{\infty} \frac{c_{m^{'}, n}}{(2 m^{'})!} \frac{2^{k} m^{'}!}{(m^{'} - k)!} p (\nabla) \nabla^{2 m^{'}} δ (x) ε^{2 m^{'} - n + 1} \\ \sim {(- 1)}^{k} \sum_{m = 0}^{\infty} \frac{p (\nabla) \nabla^{2 m} δ (x)}{W_{n, k, m}} ε^{2 m + 2 k - n + 1}, \end{array}$ and (6.2) is obtained. □

It is interesting to observe that (6.4) does not hold if $p \in P_{k}$ is not harmonic. Hence, even though (6.3) is correct for any polynomial, the simplification given by the extended Pizzetti formula will not hold unless the polynomial is harmonic.
7. Mean value theorems

There is a connection between mean value theorems for solutions of differential equations and Pizzetti’s formula, which can be seen already in [6] and later studies [17,23]. Let us consider the best known case, namely, harmonic functions. Indeed, if u is a harmonic function defined in a region Ω of $R^{n}$ then it satisfies the Mean Value Property, namely, $\begin{matrix} (7.1) & u (a) = \frac{1}{C} \int_{S} u (a + r ω) d ω \end{matrix}$ for any $a \in Ω$ if the closed ball $| x - a | ⩽ r$ is contained in Ω. Conversely, any continuous function that satisfies (7.1) for r small enough is harmonic [6,10]. This converse result follows immediately from the Pizzetti’s asymptotic formula (1.2) if u is smooth by taking $ϕ_{a} (x) = u (a + x)$ ; actually a simple argument, as we show in the proof of the Proposition 7.2 below gives the converse result if we just assume that u is locally integrable in Ω and that for each compact $K \subset Ω$ (7.1) holds for $a \in K$ and $r < r_{K}$ , where $r_{K} > 0$ .

Lemma 7.1.

Let ${r_{j}}_{j = 0}^{\infty}$ be a sequence with $r_{j} ↘ 0$ . Let $p \in H_{k}$ , where $k > 0$ . Let ϕ be a smooth function defined in a region $Ω \subset R^{n}$ . If for each $a \in Ω$ there exists $J_{a}$ such that $\begin{matrix} (7.2) & \int_{S} ϕ (a + r_{j} ω) p (ω) d ω = 0, j > J_{a}, \end{matrix}$ then in Ω, $\begin{matrix} (7.3) & p (\nabla) ϕ = 0 . \end{matrix}$

Proof.

Indeed, $\int_{S} ϕ (a + r ω) p (ω) d ω \sim C \sum_{m = 0}^{\infty} Δ^{m} p (\nabla) ϕ (a) r^{k + 2 m} / W_{n, k, m}$ as $r \to 0^{+}$ at each $a \in Ω$ , therefore (7.2) yields that all the coefficients of this asymptotic series vanish at each a, so that the $m = 0$ term gives (7.3). □

The result of the lemma might not hold for discontinuous functions, in general, but if we ask a uniformity condition on the $J_{a}$ we still obtain (7.3).

Proposition 7.2.

Let ${r_{j}}_{j = 0}^{\infty}$ be a sequence with $r_{j} ↘ 0$ . Let $p \in H_{k}$ , where $k > 0$ . Let u be a locally integrable function in a region $Ω \subset R^{n}$ . Suppose that for each compact $K \subset Ω$ there exists $J_{K} > 0$ such that for $a \in K$ we have $\begin{matrix} (7.4) & \int_{S} u (a + r_{j} ω) p (ω) d ω = 0, j > J_{K} . \end{matrix}$ Then $\begin{matrix} (7.5) & p (\nabla) u = 0, \end{matrix}$ in $D^{'} (Ω)$ .

Proof.

Notice that we may consider u as a distribution in Ω. Let $a \in Ω$ be fixed. We will show that $p (\nabla) u = 0$ in a neighborhood of a. Let $B_{0}$ be an open ball with center at a and radius $r_{0}$ such that its closure, say K, is contained in Ω. Let $η = min {r_{0}, r_{J_{K}} / 2}$ . Let ψ be any test function in $R^{n}$ with support contained in the ball $| x | ⩽ η$ , and let $ϕ = u * ψ$ in B, the ball of center a and radius η. Then ϕ is smooth in B. Furthermore, if $| b - a | < η$ then if $j > J_{K}$ we have $\begin{array}{l} \int_{S} ϕ (b + r_{j} ω) p (ω) d ω & = \int_{S} \int_{| c | < η} ψ (c) u (b + r_{j} ω - c) p (ω) d c d ω \\ = \int_{| c | < η} ψ (c) \int_{S} u (b - c + r_{j} ω) p (ω) d ω d c \\ = 0, \end{array}$ since $b - c \in K$ . It follows that $p (\nabla) ϕ = (p (\nabla) u) * ψ = 0$ in B, and this is true for any test function ψ with support contained in the ball $| x | ⩽ η$ . Taking a sequence of such test functions ${ψ_{m}}$ that converges to $δ (x)$ in $E^{'} (R^{n})$ , we obtain that $p (\nabla) u = {lim}_{m \to \infty} (p (\nabla) u) * ψ_{m}$ vanishes in B. □

Notice the simplest case, when $p = x_{l}$ .

Corollary 7.3.

Let ${r_{j}}_{j = 0}^{\infty}$ be a sequence with $r_{j} ↘ 0$ . The function u, locally integrable in Ω, does not depend on $x_{l}$ (as a distribution) if and only if for each compact $K \subset Ω$ there exists $J_{K} > 0$ such that $\begin{matrix} (7.6) & \int_{S} u (a + r_{j} ω) ω_{l} d ω = 0, j > J_{K}, \end{matrix}$ for all $a \in K$ .

Another simple yet interesting case is obtained in 2 dimensions, where we identify $R^{2}$ with $C$ , by taking $p (x, y) = x + i y$ . Proposition 7.2 yields the ensuing result that extends the classical Morera theorem [4].

Corollary 7.4.

Let ${r_{j}}_{j = 0}^{\infty}$ be a sequence with $r_{j} ↘ 0$ . A function f defined in a region $Ω \subset C$ and locally integrable there will be a.e. equal to an analytic function in Ω if and only if for each compact $K \subset Ω$ there exists $J_{K} > 0$ such that if $a \in K$ , $\begin{matrix} (7.7) & \int_{| ξ - a | = r_{j}} f (ξ) d ξ = 0, \end{matrix}$ whenever $j > J_{K}$ .

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On Pizzetti’s formula

Abstract

Keywords

1. Introduction

2. Notation

3. Radial and radial-free objects

4.1. The part of a series that is a radial multiple of a given polynomial

1 The nice proof of this fact is well known. See [18, Theorem A.9] or [10, Proposition (2.47)], for instance.

2 If p ∈ P k then ρ p ( S ) is given by (4.8), but not (4.13), that applies only if p is harmonic.

References

¹
The nice proof of this fact is well known. See [18, Theorem A.9] or [10, Proposition (2.47)], for instance.

²
If $p \in P_{k}$ then $ρ_{p} (S)$ is given by (4.8), but not (4.13), that applies only if p is harmonic.