DIVIDED DIFFERENCES AND POLYNOMIAL CONVERGENCES

Abstract

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 ∇h with size h ＞ 0, we verify that for an integer m ≥ 0 and a strictly decreasing sequence hn converging to zero, a continuous function f(x) satisfying ∇ m+1 hn f(khn) = 0, for every n ≥ 1 and k ∈ Z, turns to be a polynomial of degree ≤ m. The proof used the polynomial convergence, and additionally, we investigated several conditions on convergence to polynomials.