Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of squares of $d$ dimensional Fourier harmonics up to degree $n$.

My question is if $p$ is positive on the unit sphere $S\subset {\mathbb{R}}^n$ and can be represented as combination of spherical harmonics dimension $d$, then does there exist some spherical harmonic polynomials $g_1,\ldots, g_k$ of degree $n$ such that $p=g_1^2+\cdots g_k^2$ is a sum of squares?

i edit the question after Zach Teitler's comment.

The interval $[-\pi,\pi]^d$ means we concern the trigonometric polynomials positive on frequency domains.

The optimization problems about the polynomials positive on frequency domain $[-\pi,\pi]^d$ can be implemented via SDP approach(Gram matrix Rpresentation).

Given a positive polynomial represented as combination of spherical harmonics dimension $d$, Obviously, it is sum of squares of $d$ dimensional Fourier harmonics. Furthermore, it implies the symmetry relationship between $[-\pi,\pi] \times [0,\pi]$ and $[-\pi,\pi] \times [-\pi,0]$ on 2-sphere as an example. May be there is less information on sphere than cube? So, is it the sum of squares of spherical harmonics?

**edit after Zach Teitler's answer**

My appoligize, the theroy of positive polynomial on $[-\pi,\pi]^d$ is from
Dumitrescu, “Trigonometric Polynomials Positive on Frequency Domains and Applications to 2-D FIR Filter Design,” IEEE Transactions on Signal Processing, 2006 **Theorem 1**. That is

Given $z=[z_1,\ldots, z_d]$ and $z^k=z_1^{k_1}z_2^{k_2}\ldots z_d^{k_d}$, a Hermitian trigonommetric polynomial of degree $n$ that

$R(z)=\sum_{k=-n}^n r_k z^{-k}, r_{-k}=r_k^*$

is positive on unit $d$-circle, i.e. $z_i=e^{j\theta_i}, \theta_i \in[-\pi,\pi]^d$ , then it is sum of squares.

My question is on the relationship between sum of suqare polynomial and the polynomial that sum of squares spherical harmonics.

Thank you very much again!

Please feel free to provide any advices. Any comments and references (in English) will also be very welcome !

Thank you very much in advance!