Abstract :
P. Sablonnière introduced the so-called left Bernstein quasi-interpolant, and proved that the sequence of the approximating polynomials converges pointwise in high-order rate to each sufficiently smooth approximated function. On the other hand, Z.-C. Wu proved that the sequence of the norms of the operators is bounded. In this paper, we extract the essence why Sablonnièreʹs operator exhibits good convergence and stability properties, and we clarify a sufficient condition for general operators to have similar properties. Moreover, regarding the family of the general operators, we derive detailed results about the derivatives of the approximating polynomials that estimate their uniform convergence degree, using a convenient differentiability condition on approximated functions. Our results readily imply all the preceding ones.
