---
tag: Q6KCF4688J
kind: statement
role: theorem
name: Korovkin's theorem
msc: [41A36, 41A35]
uses: [AQDXWH4812, 08FPABDQ4C, 0Q4DR37DVE]
about: [AQDXWH4812, XAARVNKXF1]
sources:
  - url: https://en.wikipedia.org/wiki/Korovkin_approximation
same_as:
  wikidata: Q1784675
  wikipedia: Korovkin approximation
---

## Statement

Let $a < b$ and let $C[a, b]$ be the space of continuous functions $[a, b] \to \R$ with the supremum norm.
Let $L_n \colon C[a, b] \to C[a, b]$, $n \ge 1$, be [[positive linear operators|AQDXWH4812]]: each $L_n$ is linear, and $L_n f \ge 0$ whenever $f \ge 0$.
Let $e_0(x) = 1$, $e_1(x) = x$ and $e_2(x) = x^2$.
If $L_n e_i \to e_i$ [[uniformly|08FPABDQ4C]] on $[a, b]$ for $i = 0, 1, 2$, then $L_n f \to f$ uniformly on $[a, b]$ for every $f \in C[a, b]$.

## Remarks

The idea of the proof: fix $f$ and $\varepsilon > 0$.
By [[uniform continuity|0Q4DR37DVE]] there is $\delta > 0$ with $\abs{f(t) - f(x)} \le \varepsilon + 2 \norm{f}_\infty (t - x)^2 / \delta^2$ for all $t, x \in [a, b]$.
Apply the positive operator $L_n$ in the variable $t$ and evaluate at $x$.
This bounds $\abs{L_n f(x) - f(x)}$ by $\varepsilon$ plus terms in $L_n e_0 - e_0$, $L_n e_1 - e_1$ and $L_n e_2 - e_2$, which tend to $0$ uniformly.

Example: the Bernstein operators $B_n$ on $[0, 1]$ satisfy $B_n e_0 = e_0$, $B_n e_1 = e_1$ and $B_n e_2 = e_2 + x(1 - x)/n$.
So Korovkin's theorem gives $B_n f \to f$ uniformly for every continuous $f$.

Positivity is needed.
In the periodic setting the test functions are $1, \cos x, \sin x$.
The partial sums of Fourier series reproduce these test functions for $n \ge 1$, but they are not positive operators.
And there are continuous periodic functions whose Fourier series do not converge uniformly.

## Explanations

### L1 human

Many approximation schemes are positive linear operators: they average values of $f$ with nonnegative weights.
Korovkin's theorem says that to check such a scheme works for every continuous function, it is enough to check three functions: $1$, $x$ and $x^2$.
This turns a hard question into a short computation.

### L2 human

The theorem is the standard tool to prove convergence of the Bernstein, Szász–Mirakyan, Baskakov and Kantorovich operators and of Fejér means.
The periodic version uses the test functions $1, \cos x, \sin x$, and it proves Fejér's theorem on Cesàro means of Fourier series.
Quantitative forms bound $\norm{L_n f - f}_\infty$ by the modulus of continuity of $f$ and the errors on the test functions.
The theorem also shows a rigidity: a single positive linear operator on $C[a, b]$ that fixes $1$, $x$ and $x^2$ must be the identity.