Weighted polynomial approximation

Remarks

The decay condition on $w$ ensures that every polynomial lies in $C_w$, so $E_n(f)_w$ is finite. $E_n(f)_w$ is the error of best approximation of $f$ from $\mathcal{P}_n$ in the seminorm $\norm{\cdot}_w$. Polynomials are dense in $C_w$ exactly when $E_n(f)_w \to 0$ for every $f \in C_w$; Bernstein's approximation problem asks for which weights this holds. The same name is used for the weighted $L^p$ error $\inf_p \left( \int_\R \abs{f - p}^p w \, dx \right)^{1/p}$, and for weights on an interval. Example: $w(x) = e^{-x^2}$ satisfies the decay condition, since $x^k e^{-x^2} \to 0$ for each $k$. The function $f(x) = \abs{x}$ is continuous with $\abs{x} e^{-x^2} \to 0$, so $f \in C_w$, and $E_n(f)_w \le \norm{f}_w = \sup_x \abs{x} e^{-x^2} = e^{-1/2}/\sqrt{2}$ for every $n$, using $p = 0$. In the weighted $L^2$ version with the same weight, the best constant approximation of $x^2$ is $c = \int x^2 e^{-x^2} dx \big/ \int e^{-x^2} dx = 1/2$. Non-example: $w \equiv 1$ fails the decay condition, and then $\norm{p}_w = \infty$ for every nonconstant polynomial $p$.