Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a subalgebra of $C(X;\mathbb{R})$.
The Stone-Weierstrass theorem asserts that if $\mathcal{A}$ contains the constants and separates the points of $X$, then $\mathcal{A}$ is dense in $(C(X;\mathbb{R}), \|\cdot\|_\infty)$.
Suppose now that $\mathcal{A}$ is countably spanned, that is: $\mathcal{A}=\mathrm{span}_{\mathbb{R}}(\xi_\alpha\mid \alpha\in I)$ for $(\xi_\alpha)_{\alpha\in I}=:\xi$ some countable linearly independent family in $C(X;\mathbb{R})$. (So in particular, $\xi$ is such that $\xi_\alpha\cdot\xi_{\tilde{\alpha}}\in\mathrm{span}(\xi)$ for any $\alpha, \tilde{\alpha}\in I$).
As a vector space, this $\mathcal{A}$ can be normed via $\|\sum_{\alpha\in J} c_\alpha\xi_\alpha\|:= (\sum_{\alpha\in J} c_\alpha^2)^{1/2}$ (any finite $J\subseteq I$).
Assuming that $\xi$ contains a constant and separates points, we for each $f\in C(X;\mathbb{R})$ can find a sequence $(p_k)_{k\in\mathbb{N}}$ in $\mathcal{A}$ such that
$$\tag{1}\|f – p_k\|_\infty \,\leq \, \tfrac{1}{k} \quad \text{for each } \ k\in\mathbb{N}.$$
My question: Do we have any reason to believe that the associated sequence of [coefficient-]norms $(\|p_k\|)_{k\in\mathbb{N}}$ is bounded? If not, are there some extra conditions under which boundedness would hold?
(Extra conditions might be that $\xi$ is square-summable, i.e. $\|\xi(x)\|_{L^2}:=\sum_{\alpha\in I}\xi_\alpha(x)^2 < \infty$ for each $x\in X$, etc.)
Best Answer
The answer is negative. Suppose we approximate a continuous function on $[-1,1]$ with ordinary polynomialss $P_n$. If the coefficients are bounded, say $|a_{n,k}|\leq C$, then $$|P_n(z)|\leq C(1-|z|)^{-1},\quad |z|<1$$ Therefore $\{ P_n\}$ is a normal family in the unit disk, and thus there is a subsequence which converges to an analytic function uniformly on compact subsets of $|z|<1$. So if the function which we approximate is not analytic, the coefficients cannot be bounded.
This also suggests a sufficient condition for boundedness of coefficients: if the function $f$ that you approximate is analytic in a disk $|z|<1+\epsilon,$ with some $\epsilon>0$, then its Taylor coefficients are bounded by Cauchy inequality, and initial segments of the Taylor series give a uniform approximation on $[-1,1]$ by polynomials with bounded coefficients.