I am trying to construct Harmonic polynomials on the sphere. What about the examples:
- $f = x^2 + y^2 + z^2$
- $g = x^4 + y^4 + z^4$
We have that $\nabla^2 f \neq 0$ by compuing the second derivative of each coordinate separately:
$$ \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) (x^4 + y^4 + z^4) = 12(x^2 + y^2 + z^2) \neq 0 $$ We have a decomposition $L^2(S^2) = \bigoplus H_d $ with $H_d = \{ \text{harmonic polynomials in }x,y,z \text{ of degree }d \}$. I am pretty sure the two functions I have constructed are in $L^2$.
$$ \int_{S^2} |f|^2 < 9 \int_{S^2} 1\, dS = 9 \times (4\pi) $$ and a similar story for $g$. How can I get an expansion of $f$ and $g$ in terms of harmonic polynomials? Are there terms of lower degree missing?
You can decompose a function on $\mathbb{S}^2$ into spherical harmonics, not into harmonic polynomials. They are related, but they are not the same. Spherical harmonics satisfy the equation $$ \Delta_{\mathbb{S}^2} Y^m_l(\theta, \phi) = -l(l+1) Y^m_l (\theta, \phi)$$ which means that functions $f(x,y,z) = r^l Y^m_l(\theta,\phi)$ are harmonic functions. Such functions also happen to be polynomials, so they are called harmonic polynomials. However, if you multiply a spherical harmonic with wrong power of $r$, you don't get a harmonic polynomial.
If you want to write your functions into spherical harmonic, you need to separate their radial and spherical part. In case of $f$ it's simple $$ f(x,y,z) = r^2 = \sqrt{4\pi}\, r^2 Y^0_0(\theta, \phi)$$ You can see that it involves a spherical harmonic with $l=0$ multiplied by $r^2$. The power of $r$ doesn't match, so no wonder you don't get a harmonic polynomial, $\Delta f \neq 0$.
With $g$ you have \begin{align} g(x,y,z) &= r^4 (\sin^4\theta \cos^4\phi + \sin^4\theta\sin^4\phi + \cos^4\theta) = \\ &= r^4 \left(\sin^4\theta \Big(\frac18 e^{4i\phi} + \frac34 + \frac18 e^{-4i\phi}\Big) + \cos^4\theta\right) = \\ &= r^4 \left(\sqrt{\frac{8\pi}{315}}Y^4_4(\theta,\phi) + \sqrt{\frac{8\pi}{315}}Y^{-4}_4(\theta,\phi)+ \frac34 \sin^4\theta + \cos^4\theta\right) = \\ &= r^4 \left(\sqrt{\frac{8\pi}{315}}Y^4_4(\theta,\phi) + \sqrt{\frac{8\pi}{315}}Y^{-4}_4(\theta,\phi)+ \frac74 \cos^4\theta - \frac32 \cos^2\theta +\frac34\right) = \\ &= r^4 \left(\sqrt{\frac{8\pi}{315}}Y^4_4(\theta,\phi) + \sqrt{\frac{8\pi}{315}}Y^{-4}_4(\theta,\phi)+ \frac{4\sqrt\pi}{15} Y^0_4(\theta,\phi) +\frac{6\sqrt\pi}{5}Y^0_0(\theta,\phi)\right)\end{align} Again, you have spherical harmonic with index $l=0$ not matching the power of $r$, so you don't get a harmonic polynomial, and $\Delta g \neq 0$.