# Sum of Squares and Harmonic Functions

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: $$$$\boxed{\text{Is there any sum of squares } p(x) \text{ such that } p_{2i}(e_1)=a_i \text{ for } i=0,\dots, k?}$$$$ Here $e_1=(1,0,\dots,0)\in \mathbb{R}^d$. If so, how can I found one of those?

Thanks!

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