This question may look trivial, but I wonder of any results related to my question.

I consider the following polynomial: \begin{equation} p(\omega) = p_k(\omega) + Y_2(\omega) r_k(\omega), \, w\in \mathbb{R}^2, \, k\geq 1, \end{equation} where $ p_k, \, r_k$ are real-valued homogeneous polynomials of degree $ k$ , $ Y_2(\omega)$ is a harmonic polynomial (which is then homogeneous) of degree 2.

I know the values of $ p$ for all $ \omega\in \mathbb{S}^1$ (note that $ |\omega|^2 = \omega_1^2 + \omega_2^2 = 1$ on $ \mathbb{S}^1$ ); polynomials $ p_k, \, r_k$ are unknown. But I would like to reconstruct info as much as I could about them. For example I know that $ |r_k| << |p_k|$ on $ \mathbb{S}^1$ .

I know that reconstruction of $ p_k, \, r_k$ cannot be done in general, if the values are given only on $ \mathbb{S}^1$ . But I wonder what can one say bout this problem in general, for any reasonable assumptions.