DIVIDED DIFFERENCES AND POLYNOMIAL CONVERGENCES

The continuous analysis, such as smoothness and uniform convergence, for polynomials and polynomial-like functions using differential operators have been studied considerably, parallel to the study of discrete analysis for these functions, using difference operators. In this work, for the difference operator ${\nabla}_h$ with size h > 0, we verify that for an integer $m{\geq}0$ and a strictly decreasing sequence $h_n$ converging to zero, a continuous function f(x) satisfying $${\nabla}_{h_n}^{m+1}f(kh_n)=0,\text{ for every }n{\geq}1\text{ and }k{\in}{\mathbb{Z}}$$, turns to be a polynomial of degree ${\leq}m$. The proof used the polynomial convergence, and additionally, we investigated several conditions on convergence to polynomials.