Let \(H_d\subset\mathbb C[x,y]\) be the space of harmonic polynomials homogeneous of degree \(d\).
- Compute \(\dim H_d\).
- Give an explicit nonzero example \[ p(x,y)=\sum a_{ij}x^iy^j\in H_d. \]
- Show that the restrictions \[ \bigoplus_d H_d\big|_{S^1} \] span a dense subset of \(C(S^1)\) with the supremum norm.
- Explain these results in terms of analytic functions.
Solution
For \(d=0\) one has \(H_0=\mathbb C\), so \(\dim H_0=1\). For every \(d\ge1\), \[ H_d=\operatorname{span}_{\mathbb C}\{(x+iy)^d,(x-iy)^d\}, \qquad \dim H_d=2. \] Thus \((x+iy)^d\) is an explicit nonzero element of \(H_d\). On \(S^1\), these two basis elements restrict to \(e^{id\theta}\) and \(e^{-id\theta}\). Hence the restrictions of \(\bigoplus_d H_d\) are precisely the trigonometric polynomials, which are dense in \(C(S^1)\) in the supremum norm.
1. Introduce the complex coordinates\[ z=x+iy, \qquad \overline z=x-iy. \]Then every homogeneous polynomial \(p\in\mathbb C[x,y]\) of degree \(d\) canbe written uniquely as\[ p(z,\overline z)=\sum_{k=0}^d c_k z^k\overline z^{\,d-k}. \]
The linear change of variables \[ x=\frac{z+\overline z}{2}, \qquad y=\frac{z-\overline z}{2i} \] is invertible over \(\mathbb C\). Therefore \(\mathbb C[x,y]=\mathbb C[z,\overline z]\), and homogeneity of degree \(d\) gives the displayed basis.
2. For \(d\ge1\), a homogeneous polynomial \(p\) is harmonic if and only if\[ p(z,\overline z)=a z^d+b\overline z^{\,d} \]for some \(a,b\in\mathbb C\).
Using the Wirtinger operators, \[ \Delta=\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} =4\frac{\partial}{\partial z} \frac{\partial}{\partial\overline z}. \] Thus for \[ p=\sum_{k=0}^d c_k z^k\overline z^{\,d-k} \] one has \[ \Delta p =4\sum_{k=1}^{d-1} c_k k(d-k)z^{k-1}\overline z^{\,d-k-1}. \] The monomials occurring here are linearly independent. Therefore \(\Delta p=0\) exactly when \[ c_k=0 \qquad(1\le k\le d-1). \] The two surviving terms are \(c_dz^d\) and \(c_0\overline z^{\,d}\). Conversely, both are harmonic because each is killed by one of the two Wirtinger derivatives.
3. Consequently\[ \dim H_0=1, \qquad \dim H_d=2\quad(d\ge1), \]and one explicit nonzero element of \(H_d\) is\[ p_d(x,y)=(x+iy)^d. \]
For \(d=0\), the homogeneous polynomials are the constants. For \(d\ge1\), 2 gives the basis \[ \{z^d,\overline z^{\,d}\}. \] These two polynomials are linearly independent, since they are distinct monomials in the coordinates \(z,\overline z\). The stated dimension and example follow.
4. The restrictions of the spaces \(H_d\) to the unit circle span exactlythe trigonometric polynomials.
Write \(z=e^{i\theta}\) on \(S^1\). Then \[ z^d=e^{id\theta}, \qquad \overline z^{\,d}=e^{-id\theta}. \] Hence finite sums of restrictions of homogeneous harmonic polynomials are exactly the functions \[ \sum_{k=-N}^N a_k e^{ik\theta}, \] that is, the trigonometric polynomials.
5. The space\[ \left.\bigoplus_{d\ge0}H_d\right|_{S^1} \]is dense in \(C(S^1)\) in the supremum norm.
By 4 it is the algebra of trigonometric polynomials. This algebra contains the constants, is closed under complex conjugation, and separates points of \(S^1\) because the coordinate function \(z\mapsto z\) belongs to it. The complex Stone–Weierstrass theorem therefore implies that its uniform closure is all of \(C(S^1)\).
6. In analytic-function language, homogeneous harmonic polynomials arethe degree-\(d\) holomorphic and antiholomorphic modes.
The function \(z^d\) is holomorphic and \(\overline z^{\,d}\) is antiholomorphic, so 2 says every complex-valued homogeneous harmonic polynomial is a linear combination of one holomorphic and one antiholomorphic monomial. For real-valued harmonic polynomials this becomes the familiar basis \[ \operatorname{Re}(z^d), \qquad \operatorname{Im}(z^d). \] Their boundary values are the cosine and sine modes. Thus the density statement is exactly the density of finite Fourier sums on the circle; the corresponding harmonic polynomials are their harmonic extensions into the disk.