The sharpness of Lorentz’s theorem on incomplete polynomials

Abstract:

For any fixed $\theta$ with $0 < \theta < 1$, G. G. Lorentz recently showed that bounded sequences $\{\Sigma _{\theta {n_i} \leqslant k \leqslant {n_i}} {{a_k}(i){{(1 + t)}^k}\} _{i = 1}^\infty }$ of incomplete polynomials on $[ - 1, + 1]$ tend uniformly to zero on closed intervals of $[ - 1,\Delta (\theta ))$, where $2{\theta ^2} - 1 \leqslant \Delta (\theta ) < 2\theta - 1$. In this paper, we show that $\Delta (\theta ) = 2{\theta ^2} - 1$ is best possible, and that the geometric convergence to zero of such sequences on closed intervals $[{t_0},{t_1}]$ can be precisely bounded above as a function of ${t_j}$ and $\theta$. Extensions of these results to the complex plane are also included.