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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:

$$
left(
fracpartial^2partial x^2 +
fracpartial^2partial y^2 +
fracpartial^2partial z^2
right) (x^4 + y^4 + z^4) = 12(x^2 + y^2 + z^2) neq 0
$$
We have a decomposition $L^2(S^2) = bigoplus H_d $ with $H_d = textharmonic polynomials in x,y,z text of degree d $. I am pretty sure the two functions I have constructed are in $L^2$.

$$ int_S^2 |f|^2 < 9 int_S^2 1, dS = 9 times (4pi) $$
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 $mathbbS^2$ into spherical harmonics, not into harmonic polynomials. They are related, but they are not the same. Spherical harmonics satisfy the equation
$$ Delta_mathbbS^2 Y^m_l(theta, phi) = -l(l+1) Y^m_l (theta, phi)$$
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
$$ f(x,y,z) = r^2 = sqrt4pi, r^2 Y^0_0(theta, phi)$$
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
beginalign g(x,y,z) &= r^4 (sin^4theta cos^4phi + sin^4thetasin^4phi + cos^4theta) = \
&= r^4 left(sin^4theta Big(frac18 e^4iphi + frac34 + frac18 e^-4iphiBig) + cos^4thetaright) = \
&= r^4 left(sqrtfrac8pi315Y^4_4(theta,phi) + sqrtfrac8pi315Y^-4_4(theta,phi)+ frac34 sin^4theta + cos^4thetaright) = \
&= r^4 left(sqrtfrac8pi315Y^4_4(theta,phi) + sqrtfrac8pi315Y^-4_4(theta,phi)+ frac74 cos^4theta - frac32 cos^2theta +frac34right) = \
&= r^4 left(sqrtfrac8pi315Y^4_4(theta,phi) + sqrtfrac8pi315Y^-4_4(theta,phi)+ frac4sqrtpi15 Y^0_4(theta,phi) +frac6sqrtpi5Y^0_0(theta,phi)right)endalign
Again, you have spherical harmonic with index $l=0$ not matching the power of $r$, so you don't get a harmonic polynomial, and $Delta g neq 0$.