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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.
41A10 Approximation by polynomials
39A10 Additive difference equations
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