# Relation between error and coefficients in polynomial approximation

Let $f$ be a function and $(p_i)_{i=0}^\infty$ a sequence of polynomials such that the first $n$ span the space $\mathcal{P}_n$ of polynomials of degree $\leq n$ for all $n$. Furthermore, let $$\DeclareMathOperator*{\argmin}{arg\,min} c^{(n)} := \argmin_{c^{(n)} \in \mathbb{C}^n} \|f - \sum_{i = 0}^n c^{(n)}_i p_i\|_\infty .$$ Assuming I know $e_n := \min_{p_n \in \mathcal{P}_n} \|f - p_n\|_\infty$, can I then say something about the decay behaviour of $c := \lim_{n \to \infty} c^{(n)}$? For example, if $e_n \propto \exp(-\gamma n)$, I would intuitively expect $c_i \propto \exp(-\gamma n)$ as well.

The behaviour of the error tells you nothing about the decay in the coefficients. Consider a basis $$p_k$$ constructed as follows:$$\DeclareMathOperator*{\argmin}{arg\,min}$$ $$p_k := \begin{cases} \argmin\limits_{p \in \mathcal{P}_k} \|f - p\|_\infty & \text{if } e_k < e_{k-1}, \\ x^k & \text{otherwise}. \end{cases}$$ Then, $$c^{(n)}$$ is one at the last position $$k$$ such that $$e_k < e_{k-1}$$ and zero otherwise. Assuming $$e_k > 0$$ for all finite $$k$$ (i.e. $$f$$ is not a polynomial), then $$c^{(n)}$$ has an entry of magnitude 1 arbitrarily far out for $$n$$ large enough and hence does not decay. In fact, it doesn't even converge.