The eigenstructure of the Bernstein operator. (English) Zbl 0963.41006

Cooper, Shaun; Waldron, Shayne

J. Approximation Theory 105, No. 1, 133-165 (2000).

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.

Reviewer: Dany Leviatan (Tel Aviv)

MSC:

41A10	Approximation by polynomials
41A36	Approximation by positive operators

Keywords:

eigenvalues and eigenfunctions of the Bernstein operators

References:

[1]	Ando, T., Totally positive matrices, Linear Algebra Appl., 90, 165-219 (1987) · Zbl 0613.15014
[2]	Berens, H.; DeVore, R., A characterisation of Bernstein polynomials, (Cheney, E. W., Approximation Theory III (1980), Academic Press: Academic Press New York), 213-219 · Zbl 0483.41016
[3]	Berens, H.; Xu, Y., On Bernstein-Durrmeyer polynomials with Jacobi weights, (Chui, C. K., Approximation Theory and Functional Analysis (1991), Academic Press: Academic Press Boston), 25-46 · Zbl 0715.41013
[4]	Butzer, P. L., Linear combinations of Bernstein polynomials, Canad. J. Math., 5, 559-567 (1953) · Zbl 0051.05002
[6]	Derriennic, M. M., Sur l’approximation de fonctions intégrables sur [0, 1] par des polynômes de Bernstein modifiés, J. Approx. Theory, 31, 325-343 (1981) · Zbl 0475.41025
[7]	Ditzian, Z.; Totik, V., Moduli of Smoothness (1987), Springer-Verlag: Springer-Verlag New York · Zbl 0666.41001
[8]	Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0052.29502
[9]	Gonska, H. H.; Long Zhou, Xin, Approximation theorems for the iterated Boolean sums of Bernstein operators, J. Comput. Appl. Math., 53, 21-31 (1994) · Zbl 0816.41020
[10]	Karlin, S., Total Positivity (1968), Stanford Univ. Press: Stanford Univ. Press Stanford
[11]	Karlin, S.; Ziegler, Z., Iteration of positive approximation operators, J. Approx. Theory, 3, 310-339 (1970) · Zbl 0199.44702
[12]	Kelisky, R. P.; Rivlin, T. J., Iterates of Bernstein polynomials, Pacific J. Math., 21, 511-520 (1967) · Zbl 0177.31302
[13]	Lorentz, G. G., Bernstein Polynomials (1953), Toronto Press: Toronto Press Toronto
[14]	Micchelli, C., The saturation class and iterates of the Bernstein polynomials, J. Approx. Theory, 8, 1-18 (1973) · Zbl 0258.41012
[15]	Pinkus, A., Spectral properties of totally positive kernels and matrices, (Gasca, M.; Micchelli, C. A., Total Positivity and Its Applications (1996), Kluwer Academic: Kluwer Academic Dordrecht), 477-511 · Zbl 0891.45001
[16]	Sablonnière, P., A family of Bernstein quasi-interpolants on [0, 1], Approx. Theory Appl., 8, 62-76 (1992) · Zbl 0772.41024
[17]	Sevy, J. C., Lagrange and least-squares polynomials as limits of linear combinations of iterates of Bernstein and Durrmeyer polynomials, J. Approx. Theory, 80, 267-271 (1995) · Zbl 0823.41022
[18]	da Silva, M. R., Nonnegative order iterates of Bernstein polynomials and their limiting semigroup, Portugal. Math., 42, 225-248 (1985) · Zbl 0562.41007
[19]	Wenz, H.-J., On the limits of (linear combinations of) iterates of linear operators, J. Approx. Theory, 89, 219-237 (1997) · Zbl 0871.41014
[20]	Widder, D. V., The Laplace Transform (1941), Princeton University Press: Princeton University Press Princeton · Zbl 0060.24801
[21]	Zhou, D. X., On smoothness characterized by Bernstein type operators, J. Approx. Theory, 81, 303-315 (1995) · Zbl 0849.41014