Spherical harmonics

Let the function $u$ on $\mathbb {R}^{2}$ be harmonic, not identically $0$, and homogeneous of degree $d$, where $d > 0$.

To say that $u$ is homogeneous of degree $d$ means that $u(tx,ty) = t^d u(x,y)$ for $t > 0$.

Prove that $d$ is an integer.

Fix a positive integer $n$. The Laplacian $\Delta p$ of a twice differentiable realvalued function $p$ on $\mathbb {R}^{n}$ is the function on $\mathbb {R}^{n}$ defined by

The function $p$ is called harmonic if $\Delta p = 0$. A polynomial on $\mathbb {R}^{n}$ is a linear combination (with coefficients in $\mathbb {R}$) of functions of the form $x_{1}^{m_{1}} \ldots x_{n}^{m_{n}}$, where $m_{1}, \ldots , m_{n}$ are nonnegative integers.

Suppose $q$ is a polynomial on $\mathbb {R}^{n}$. Prove that there exists a harmonic polynomial $p$ on $\mathbb {R}^{n}$ such that $p(x) = q(x)$ for every $x \in \mathbb {R}^{n}$ with $\left\| x\right\| = 1$.

Construct $2 m+1$ linearly independent homogeneous harmonic polynomials of degree $m$.

Show that every homogeneous polynomial $f \in P_{m}$ can be written as a linear combination, with coefficients depending on $x^{2}+y^{2}+z^{2}$, of harmonic polynomials of degrees $m, m-2, m-4, \ldots$

Deduce from Exercise 2 that every polynomial function $\tilde{g}:(X, Y, Z) \mapsto g(x, y, z)$ on the sphere $S^{2}: x^{2}+y^{2}+z^{2}=1$ expands in spherical functions — the restrictions of harmonic polynomials to $S^{2}$.

The parity operator $\Pi$ is defined by

Let $Y_{\ell m}$ be the spherical harmonics, where $\ell = 0,1,\ldots$ and $m = -\ell ,-\ell +1,\ldots ,+\ell$. Find $\Pi Y_{\ell m}$.