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Universal properties of approximation operators

A. Vogt

We discuss universal properties of some operators $L_n:C[0,1]\rightarrow C[0,1].$ The operators considered are closely related to a theorem of Korovkin [Kor59] which states that a sequence of positive linear operators $L_n$ on $C[0,1]$ is an approximation process if $L_n f_i \rightarrow f_i\,\,(n\to\infty)$ uniformly for $i=0,1,2$, where $f_i(x)=x^i$. We show that $L_n f$ may diverge in a maximal way if any requirement concerning $L_n$ in this theorem is removed. There exists for example a continuous function $f$ such that $(L_nf)_{n\in\mathbb{N}}$ is dense in $(C[0,1],\left\|.\right\|_\infty)$, even if $L_n$ is positive, linear and satisfies $L_n P\rightarrow P\,\,(n\to\infty)$ for all polynomials $P$ with $P(0)=0$.