Korovkin's Theorem

In mathematics when and how a given series of functions converges to a given function is a basic question, for example, the fourier series, the power series and so on, these problem can be view as a kind of approximation.

As our first choice, I would like to share the Korovkin's Theorem.
Recall that $C([0,1])$ is the space of continuous functions on $[0,1]$, which is a linear space, thus we can define the linear mapping between $C[0,1]$ and $C[0,1]$, also called (linear) operators on $C[0,1]$. We shall call a operator $F$ on $C[0,1]$ be positive if $$F(f)\geq0,\quad \forall f\geq0, f\in C[0,1].$$

Let ${F_n}_{n=1}^\infty$ be a sequence of operators on $C[0,1]$, then the Korovkin's Theorem states that $F_n(f)\rightrightarrows f$ (uniformly on $[0,1]$), $\forall f\in C[0,1]$, provided that $F_n(e_i)\rightrightarrows e_i$, $e_i=t^i$, $t\in[0,1]$, $i=0,1,2$.

Thus, we only need to verify it for three special functions (we call it as good basis of $C[0,1]$) $e_i$ to show that $F_n(f)$ is uniformly convergent to $f$ for any $f\in C[0,1]$.

As an application, we can easily show that the Bernstein polynomial uniformly convergence to $f$, for any $f\in C[0,1]$. See Morten's Notes for detail.

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Reference

1. Korovkin, P. P. "On convergence of linear positive operators in the space of continuous functions."Dokl. Akad. Nauk SSSR. Vol. 90. 1953.
2. A Note on Korovkin’s Theorem by Morten Nielsen

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