Saturation of approximation methods

Remarks

Example: the Bernstein operators $B_n$ on $C[0, 1]$ with the uniform norm are saturated with order $1/n$. For $f(x) = x^2$ one has $B_n f(x) = x^2 + \frac{x(1 - x)}{n}$, so $\norm{B_n f - f} = \frac{1}{4n}$, which is $O(1/n)$ but not $o(1/n)$. The only $f$ with $\norm{B_n f - f} = o(1/n)$ are the linear functions, which $B_n$ reproduces.

Non-example: the Fourier partial sum operators $S_n$ on $L^2$ of the circle are not saturated for any order. A trigonometric polynomial $f$ of degree $k \ge 1$ has $S_n f = f$ for $n \ge k$, so $\norm{S_n f - f} = o(\varphi(n))$, but $S_0 f \ne f$, so clause 1 fails.

The Fejér means of Fourier series are saturated with order $1/n$ in $C$ of the circle, and only constants are approximated with error $o(1/n)$. Saturation theorems typically identify the saturation class with a smoothness class, such as functions with Lipschitz first derivative for Bernstein operators.