Harmonic Polynomial

Harmonic Polynomial

Recall the Dirichlet problem for the ball in \(\mathbf R^n\): given \(f\in C(S)\), find \(u\in C(\overline{B})\) such that \(u\) is harmonic on \(B\) and \(u|_S = f\). We know from Chapter 1 that

\[u(x)=P[f](x)=\int_S f(\zeta)\,\frac{1-|x|^2}{|x-\zeta|^n}\,d\sigma(\zeta) \]

for \(x\in B\). To prove that \(P[f]\) is harmonic on \(B\), we computed its Laplacian by differentiating under the integral sign in the equation above and noting that for each fixed \(\zeta\in S\), the Poisson kernel \((1-|x|^2)/|x-\zeta|^n\) is harmonic as a function of \(x\).

Suppose now that \(f\) is a polynomial on \(\mathbf R^n\) restricted to \(S\). For fixed \(\zeta\in S\), the Poisson kernel \((1-|x|^2)/|x-\zeta|^n\) is not a polynomial in \(x\), so nothing in the formula above suggests that \(P[f]\) should be a polynomial. Thus our first result in this chapter should be somewhat of a surprise: \(P[f]\) is indeed a polynomial, and its degree is at most the degree of \(f\).

Further indications of the importance of harmonic polynomials will come when we prove that every polynomial on \(\mathbf R^n\) can be written as the sum of a harmonic polynomial and a polynomial multiple of \(|x|^2\). This result will then be used to decompose the Hilbert space \(L^2(S)\) into a direct sum of spaces of harmonic polynomials. As we will see, this decomposition is the higher‑dimensional analogue of the Fourier series decomposition of a function on the unit circle in \(\mathbf R^2\).

Polynomial Decompositions

5.1 Theorem:

If \(p\) is a polynomial on \(\mathbf R^n\) of degree \(m\), then

\[P[p|_S]=(1-|x|^2)q+p \]

for some polynomial \(q\) of degree at most \(m-2\).

\(P[p|_S]\) : 边界函数 \(P|S\) 对应的泊松积分（球内调和解）。

5.3 Corollary:

No nonzero polynomial multiple of \(|x|^2\) is harmonic.

In the next section we will be working in \(L^2(S)\). Two distinct polynomials of the same degree can have equal restrictions to \(S\), but two homogeneous polynomials of the same degree that agree on \(S\) must agree everywhere. Thus we will find it convenient to restrict attention to homogeneous polynomials. Let us denote by \(\mathcal{P}_m(\mathbf R^n)\) the complex vector space of all homogeneous polynomials on \(\mathbf R^n\) of degree \(m\). Let \(\mathcal{H}_m(\mathbf R^n)\) denote the subspace of \(\mathcal{P}_m(\mathbf R^n)\) consisting of all homogeneous harmonic polynomials on \(\mathbf R^n\) of degree \(m\). For example,

5.4 \(p(x,y,z)=8x^5-40x^3y^2+15xy^4-40x^3z^2+30xy^2z^2+15xz^4\)

is an element of \(\mathcal{H}_5(\mathbf R^3)\), as the reader can verify; we have used \((x,y,z)\) in place of \((x_1,x_2,x_3)\) to denote a typical point in \(\mathbf R^3\).

这里定义了:
\(\mathcal{P}_m(\mathbf R^n)\)、\(\mathcal{H}_m(\mathbf R^n)\)

5.5

In the next section we will see that this is an orthogonal decomposition when we restrict all functions to \(S\) and use the usual inner product that comes from surface‑area measure.

Proposition: If \(m\ge 2\), then

\[\mathcal{P}_m(\mathbf R^n)=\mathcal{H}_m(\mathbf R^n)\oplus |x|^2\mathcal{P}_{m-2}(\mathbf R^n). \]

We now come to the main result of this section. As usual, \([t]\) denotes the largest integer less than or equal to \(t\). Thus in the theorem below, the last index \(m-2k\) equals \(0\) or \(1\), depending upon whether \(m\) is even or odd.

5.7 Theorem: Every \(p\in \mathcal{P}_m(\mathbf R^n)\) can be uniquely written in the form

\[p = p_m + |x|^2 p_{m-2} + \cdots + |x|^{2k} p_{m-2k}, \]

where \(k=\left[\frac{m}{2}\right]\) and each \(p_j\in \mathcal{H}_j(\mathbf R^n)\).

PROOF: The desired result obviously holds when \(m=0\) or \(m=1\), because \(\mathcal{P}_m(\mathbf R^n)=\mathcal{H}_m(\mathbf R^n)\) in those cases. Thus we can assume that \(m\ge 2\).

Suppose that \(p\in \mathcal{P}_m(\mathbf R^n)\). By the previous proposition, \(p\) can be uniquely written in the form

\[p = p_m + |x|^2 q, \]

\([t]\): denotes the largest integer less than or equal to \(t\). Thus in the theorem below, the last index \(m-2k\) equals \(0\) or \(1\), depending upon whether \(m\) is even or odd.

5.7 Theorem:

Every \(p\in \mathcal{P}_m(\mathbf R^n)\) can be uniquely written in the form

\[p = p_m + |x|^2 p_{m-2} + \cdots + |x|^{2k} p_{m-2k}, \]

where \(k=\left[\frac{m}{2}\right]\) and each \(p_j\in \mathcal{H}_j(\mathbf R^n)\)

5.8 Proposition:

If \(m\ge 2\), then

\[\dim \mathcal{H}_m(\mathbf R^n)= \binom{n+m-1}{n-1} -\binom{n+m-3}{n-1}. $$. \]

\dim\mathcal{P}_m(\mathbf R^n)=\binom{n+m-1}{n-1}.

\[ \]

\dim \mathcal{H}_m(\mathbf R^n) = \dim \mathcal{P}m(\mathbf R^n) - \dim \mathcal{P}(\mathbf R^n).

\[ ## Spherical Harmonic Decomposition of $L^2(S)$ Now we turn to ideas revolving around `orthogonal` direct sums, which means that we need to introduce an `inner product`. $\mathbb{R}^n$中的单位球面（就是这本书里的球面记号） $d\sigma$: 球面$S$上的曲面面积测度（球面Lebesgue测度），球面的面积元；积分就是在 $L^2(S,d\sigma)$: 球面关于测度$d\sigma$的平方可积函数空间 定义 \]

L^2(S,d\sigma)=\left{ f:S\to\mathbb{R},\bigg|, \int_S |f(\zeta)|^2 d\sigma(\zeta) < \infty \right}

\[ 内积定义: \]

\langle f,g\rangle=\int_S f(\zeta),g(\zeta),d\sigma(\zeta)

\[ If $p,q$ are polynomials on $\mathbf{R}^n$ and $q$ is harmonic and homogeneous with degree higher than the degree of $p$, then \]

\int_S pq,d\sigma = 0.

\[ The restriction of $\mathcal{H}_m(\mathbf{R}^n)$ to $S$ is sufficiently important to receive its own name and notation. A spherical harmonic of degree $m$ is the restriction to $S$ of an element of $\mathcal{H}_m(\mathbf{R}^n)$. The collection of all spherical harmonics of degree $m$ will be denoted by $\mathcal{H}_m(S)$; thus \]

\mathcal{H}_m(S)=\bigl{,p|_S,\big|; p\in\mathcal{H}_m(\mathbf{R}^n),\bigr}.

\[ 5.12 Theorem: \]

L^{2(S)=\bigoplus_{m=0}}\mathcal{H}_m(S).

\[ ## Inner Product of Spherical Harmonics 5.13 Lemma: If $m>0$ and $p,q\in \mathcal{H}_m(\mathbb{R}^n)$, then \]

\int_S pq,d\sigma=\frac{1}{m(n+2m-2)}\int_S \nabla p\cdot\nabla q,d\sigma.

\[ 5.14 Theorem: If $p=\sum_{\alpha} b_{\alpha} x^{\alpha}$ and $q=\sum_{\alpha} c_{\alpha} x^{\alpha}$ are harmonic polynomials on $\mathbb{R}^n$, then \]

\langle p,q\rangle=\sum_{\alpha} b_{\alpha}\overline{c_{\alpha}} w_{\alpha},

\[where \]

w_{\alpha}=\frac{\alpha!}{n(n+2)\cdots(n+2|\alpha|-2)}.

\[ ## Spherical Harmonics via Differentiation For $p=\sum_{\alpha} c_{\alpha}x^{\alpha}$ a polynomial on $\mathbb{R}^n$, we define $p(D)$ to be the differential operator $\sum_{\alpha} c_{\alpha}D^{\alpha}$. 5.15 Lemma: If $p\in \mathcal{P}_m(\mathbb{R}^n)$, then $K\big[p(D)|x|^{2-n}\big]\in \mathcal{H}_m(\mathbb{R}^n)$. K:是kelvin transform <font color=red>定义了:</font> $p(D)$\]