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Divided differences and polynomial convergences. (English) Zbl 1381.41005
Summary: 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 hn converging to zero, a continuous function $$f(x)$$ satisfying $\nabla^{m+1}_{h_n}f(kh_n)=0,\text{ for every }n\geq 1\text{ and }k\in\mathbb{Z},$ 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.
##### MSC:
 41A10 Approximation by polynomials 39A10 Additive difference equations
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