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The eigenstructure of the Bernstein operator. (English) Zbl 0963.41006
The authors determine the eigenvalues and eigenfunctions of the Bernstein operator $B_n$, the latter are, of course, $n+1$ polynomials of degrees $k=0,\dots,n$. They show that the $k$th eigen-polynomial $p^{(n)}_k$ has $k$ simple zeros in $[0,1]$ and describe $\lim_{n\to\infty}p^{(n)}_k$, for fixed $k$. Applications are given to iterates of the Bernstein operators, and to Bernstein quasi-interpolants.

41A10Approximation by polynomials
41A36Approximation by positive operators
Full Text: DOI
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