On interpolation series related to the Abel-Goncharov problem (Q5947106)

For a function \(f\) defined on \([0,1]\), let NEWLINE\[NEWLINE\Delta_n f={n^n \over n!} \sum_{k=0}^n (-1)^{n-k}{n \choose k} f(k/n),NEWLINE\]NEWLINE and let \(p_n\) be a polynomial satisfying the conditions NEWLINE\[NEWLINE\Delta_k p_n= \Delta_k f, \qquad k=0,1, \ldots, n.NEWLINE\]NEWLINE The paper establishes the existence of \(p_n\) for every \(f\) and provides a description of the basic polynomials \(c_k\) in the expansion NEWLINE\[NEWLINEp_n(\cdot)= \sum_{k=0}^n c_k(\cdot) \Delta_k f. NEWLINE\]NEWLINE If \(f(x)=\sum_{n=0}^\infty a_n x^n\), with \(a_n\) satisfying, for sufficiently large \(n\) and some constant \(A>0\), the inequality \(|a_n |\leq A^n/n!\), then \(f\) can be expanded into the series NEWLINE\[NEWLINEf(x)= \sum_{n=0}^\infty c_n(x) \Delta_n f,NEWLINE\]NEWLINE converging uniformly on \([0,1]\).