The Bernstein operator $B_n$ reproduces the linear polynomials, which are therefore eigenfunctions corresponding to the eigenvalue $1$. We determine the rest of the eigenstructure of $B_n$. Its eigenvalues are $$lambda_k^{(n)}:={n!over(n-k)!}{1over n^k}, qquad k=0,1,ldots,n,$$ and the corresponding monic eigenfunctions $p_k^{(n)}$ are polynomials of degree $k$, % (with interlacing zeros) which have $k$ simple zeros in $[0,1]$. By using an explicit formula, it is shown that $p_k^{(n)}$ converges as $ntoinfty$ to a polynomial related to a Jacobi polynomial. Similarly, %for fixed $k$, the dual functionals to $p_k^{(n)}$ converge as $ntoinfty$ to measures that we identify. This diagonal form of the Bernstein operator and its limit, the identity (Weierstrass density theorem), is applied to a number of questions. These include the convergence of iterates of the Bernstein operator, and why Lagrange interpolation (at $n+1$ equally spaced points) fails to converge for all continuous functions whilst the Bernstein approximants do. We also give the eigenstructure of the Kantorovich operator. Previously, the only member of the Bernstein family for which the eigenfunctions were known explicitly was the Bernstein--Durrmeyer operator, which is self adjoint. |
Keywords
(multivariate) Bernstein operator, diagonalisation, eigenvalues, eigenfunctions, total positivity, Stirling numbers, Jacobi polynomials, semigroup, quasi--interpolant
Math Review Classification
Primary 41A10, 15A18, 38B42
; Secondary 33C45, 41A36
Last Updated
20 October 1999
Length
28 pages
Availability
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