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 $S\subset {\mathbb{R}}^n$, 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 consern 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 on represented as combination of spherical harmonics dimension $d$, it is obviously that it is sum of squares of $d$ dimensional Fourier harmonics. But, is it the sum of squares of spherical harmonics?

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

Thank you very much in advance!