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Let $a_0, a_1, \dots a_k$ be non-negative reals. Any homogeneous polynomial $p$ of degree $2k$ in $\mathbb{R}^{d}$ can be decomposed as $$ p(x)=\sum_{i=0}^{k}c_i|x|^{2(k-i)}p_{2i}(x) $$ for some $c_i$, where each $p_{2i}$ is a spherical harmonic of degree $2i$. My question is: $$ \begin{equation} \boxed{\text{Is there any sum of squares } p(x) \text{ such that } p_{2i}(e_1)=a_i \text{ for } i=0,\dots, k?} \end{equation} $$ Here $e_1=(1,0,\dots,0)\in \mathbb{R}^d$. If so, how can I found one of those?


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