# Uniform-degree polynomial estimation of smooth functions with bounded derivatives

Let $$\{f_i\}_{i\in I}$$ be a family of smooth functions on the circle such that, for each $$j\in\mathbb{N}$$ there exists $$C_j>0$$ such that $$\left\lVert\frac{d^j f_i}{dx^j}\right\rVert_\infty for all $$i\in I$$.

Question: Given any error $$\epsilon>0$$, can one find an integer $$N_\epsilon$$ such that for any $$i$$, there exists a polynomial $$P_i$$ of degree at most $$N_\epsilon$$ such that $$||P_i-f_i||_\infty<\epsilon$$?

Remark: It is clear that if the derivatives were not uniformly bounded in $$i$$ then this is not possible. It seems reasonable to expect that such an estimate is possible with that assumption in place, but I am looking for a rigorous proof.