Relation between error and coefficients in polynomial approximation

Question

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.

Answer 1

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.