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.


41A10 Approximation by polynomials
41A36 Approximation by positive operators
Full Text: DOI


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