# The eigenstructure of the Bernstein operator

## by Shaun Cooper and Shayne Waldron

## Abstract:

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

and_{ } the corresponding monic eigenfunctions
*p*^{(n)}_{k} are polynomials of degree *k*,
which have *k*_{ } simple^{ } zeros in [0, 1].
By using an explicit formula,_{ } it is^{ } shown that
*p*^{(n)}_{k} converges as
*n*
to a polynomial related to a^{ } Jacobi_{ } polynomial.
Similarly, the dual functionals to *p*^{(n)}_{k}
converge as
*n*
to measures that we identity.^{ } 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:**
41A10, 15A18, 38B42 (primary), 33C45, 41A36 (secondary)

**Length:**
28 pages

**Last updated:**
20 June 2000

**Status:**
*J. Approx. Theory* **105** (2000), no. 1, 133-165.

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