The Hyers-Ulam Stability Constant for Chebyshevian Bernstein Operators

Rachid Ait-Haddou (1) , Daisuke Furihata (1) , Marie-Laurence Mazure (2)

1 CMC - Cybermedia Center [Osaka]
2 CVGI - Calcul des Variations, Géométrie, Image

On a closed bounded interval, a given Extended Chebyshev space which possesses a Bernstein basis generates infinitely many operators of the Bernstein-type. We show that all these operators share the same Hyers-Ulam stability constant. This constant is the maximum, in absolute value, of the Bézier coefficients of the generalised Chebyshev polynomial associated with the given space. We establish an optimality property of these Bernstein operators with respect to the Hyers-Ulam stability constant. Numerical computations of these constants are investigated in two cases: rational and Müntz Bernstein operators, with special emphasis on their behaviour with respect to the concerned interval and to dimension elevation.

Mots clés: Bernstein-type operators; Extended Chebyshev spaces; best constant; Hyers-Ulam stability; total positivity; Bernstein bases

Domaines: Mathématiques [math]

Rachid Ait-Haddou, Daisuke Furihata, Marie-Laurence Mazure. The Hyers-Ulam Stability Constant for Chebyshevian Bernstein Operators. Journal of Mathematical Analysis and Applications, 2018, 463 (2), pp.1075-1091. ⟨10.1016/j.jmaa.2018.03.067⟩. ⟨hal-01660631⟩