and $x\in {\mathbb{R}}^n$, or equivalently if $P$ is the sum of monomials with the same degree $d$.

Set ${\mathcal{P}}_{d}$ to be the vector space of homogeneous harmonic polynomial of degree $d$ defined on ${\mathbb{R}}^n$. For $d\geq 2$ and $n\geq 3$, its dimension is given by $$\begin{gathered}
  \operatorname{dim}({\mathcal{P}}_d) = \binom{n+d-1}{n-1}-\binom{n+d-3}{n-1} \leq C(n) d^{n-2}\, .
 \end{gathered}$$

By the standard theory of spherical harmonics, recall that $$\begin{gathered}
 L^2(\partial B(0,1)) = \bigoplus_{d=0}^\infty {\mathcal{P}}_d\, ,\end{gathered}$$ where $L^2(\partial B(0,1))$ is the real Hilbert space generated by the scalar product $$\begin{gathered}
 {\left\langlef\middle\vertg\right\rangle}= \fint_{\partial B(0,1)} fg\, .\end{gathered}$$ The space ${\mathcal{H}}(B(0,1))$ inherits the Hilbert structure of $L^2(\partial B(0,1))$. Indeed, this product is well defined for all functions in $W^{1,2}(B(0,1))$, however only gives a hilbert space structure on the harmonic functions as ${\left\|u\right\|} =0 \Rightarrow u=0$ is true only on harmonic functions.

Thus, we can write $$\begin{gathered}
\label{eq_expu}
 u(y)= \sum_{d=0}^\infty a_d P_d(y)\, ,\end{gathered}$$ where $P_d$ are hhP of degree $d$ normalized with ${\left\|P_d\right\|}=1$, and $a_d = {\left\langleu\middle\vertP_d\right\rangle}$. This expansion of course will depend on the base point chosen for the expansion. When needed, we will make this dependence explicit by writing $$\begin{gathered}
 u(y+x)=\sum_{d=0}^\infty a_d(x) P_{d,x} (y)\, .\end{gathered}$$

It is clear that if $P$ is a hhP of degree $d$, then 