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.<\/p>\n
As our first choice, I would like to share the Korovkin’s Theorem<\/strong>. $$ F(f)\\geq0,\\quad \\forall f\\geq0, f\\in C[0,1]. $$<\/p>\n <\/p>\n Let ${F_n}_{n=1}^\\infty$ be a sequence of operators on $C[0,1]$, then the Korovkin’s Theorem<\/strong> states that $F_n(f)\\rightrightarrows f$ (uniformly on $[0,1]$), $\\forall f\\in C[0,1]$, provided that $F_n(e_i)\\rightrightarrows e_i$, $e_i=t^i$, $t\\in[0,1]$, $i=0,1,2$. <\/p>\n<\/div>\n Thus, we only need to verify it for three<\/strong>special<\/em> functions (we call it as good basis<\/em> of $C[0,1]$) $e_i$ to show that $F_n(f)$ is uniformly convergent to $f$ for any $f\\in C[0,1]$.<\/p>\n As an application, we can easily show that the Bernstein polynomial<\/strong> uniformly convergence to $f$, for any $f\\in C[0,1]$. See Morten’s Notes<\/a> for detail.<\/p>\n A Note on Korovkin\u2019s Theorem<\/a>by Morten Nielsen<\/em><\/p>\n
\nRecall 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<\/strong> between $C[0,1]$ and $C[0,1]$, also called (linear) operators<\/strong> on $C[0,1]$. We shall call a operator $F$ on $C[0,1]$ be positive<\/strong> if <\/p>\n
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