= Solution
Use the nonnegative <Laplacian> convention $\widetilde\Delta=-\sum_{j=1}^{d+1}\partial_j^2$ and $\Delta=-\operatorname{div}\operatorname{grad}$ on the unit <sphere>. In <polar coordinates> the Euclidean <Riemannian metric> is $dr^2+r^2g_{S^d}$, its <volume form> is $r^d\,dr\,dV_{S^d}$, and therefore the <Laplacian in polar coordinates> is
$$
\widetilde\Delta=-\partial_r^2-\frac d r\partial_r+\frac1{r^2}\Delta.
$$
Consequently the required restriction identity is
$$
\boxed{\Delta\bar f=\overline{\widetilde\Delta f}+\left.(\partial_r^2f+d\partial_rf)\right|_{r=1}.}
$$
For the opposite sign convention all <eigenvalues> below change sign, and the restriction identity changes accordingly.
Let $\mathcal P_n$ be the <vector space> of degree-$n$ <homogeneous polynomials> in $d+1$ variables, and let $\mathcal H_n$ be its subspace of <harmonic polynomials>. Write $H(r\omega)=r^nY(\omega)$ for $H\in\mathcal H_n$. Substitution in the <Laplacian> identity gives
$$
0=r^{n-2}\bigl(\Delta Y-n(n+d-1)Y\bigr).
$$
Thus each restricted <harmonic polynomial> is a <spherical harmonic> with <eigenvalue> $\lambda_n=n(n+d-1)$.
To count these <spherical harmonics>, give <polynomials> the <Fischer inner product> $\langle x^\alpha,x^\beta\rangle_F=\alpha!\delta_{\alpha\beta}$. Multiplication by $x_j$ is adjoint to $\partial_j$, so multiplication by $r^2$ is adjoint to the ordinary signed <Laplacian> $\sum_j\partial_j^2$. In these finite-dimensional <inner product spaces>, the orthogonal complement of $r^2\mathcal P_{n-2}$ is exactly $\mathcal H_n$. Hence the <Fischer decomposition> is
$$
\mathcal P_n=\mathcal H_n\oplus r^2\mathcal P_{n-2}.
$$
Multiplication by $r^2$ is injective, while restriction of a <homogeneous polynomial> to the <sphere> is injective: homogeneity recovers its value at every nonzero point. Counting <monomials> now gives the <eigenvalue multiplicity>
$$
\boxed{m_n=\dim\mathcal H_n=\binom{n+d}{d}-\binom{n+d-2}{d},\qquad n=0,1,2,\ldots,}
$$
where the second term is zero for $n<2$.
It remains to establish that these are all the <eigenvalues> and their full <multiplicities>. Iterating the <Fischer decomposition> expresses every restricted <polynomial> as a sum of the restricted <harmonic polynomials>. Restricted <polynomials> separate points and contain the constants, so the <Stone-Weierstrass theorem> makes their span dense in continuous functions on the <sphere>, hence dense in its <L2 space>. The <Laplace-Beltrami operator> is self-adjoint, and the distinct numbers $\lambda_n$ strictly increase for $d\geq1$. Its different <eigenspaces> are therefore orthogonal. Any additional <eigenfunction>, or any additional vector in one of these <eigenspaces> orthogonal to its known <spherical harmonics>, would be orthogonal to a dense subspace and hence zero. The standard compact elliptic <spectral theorem> then shows that the <spectrum of the round sphere> is precisely \b[$\lambda_n=n(n+d-1)$ with multiplicity $m_n$]. For $S^1$ this gives multiplicity one at zero and two at $n^2$ for $n\geq1$. If the degenerate case $d=0$ is included, $S^0$ has two points and its <Laplacian> is zero on a two-dimensional function space.
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