Polynomial two-point expansions
A polynomial two-point expansion of a function \(f\) is a series representation of \(f\) using data of the form \(f^{(j_ i)}(x_ i)\) where \(x_ i\in \{a,b\}\), \(i=1,2,..\). The most classical example is the Lidstone series which in terms of incidence schemes solves the interpolation problem given by the iteration of the scheme \(10\choose 10\). By a theorem of Widder, a sufficient condition on \(f\) to be represented by its Lidstone series is that \(f\) be entire and \(f^{(n)}(0)=o(\pi^ n)\) [\textit{D. V. Widder}, Trans. Am. Soc. 51, 387-398 (1942; Zbl 0027.39202)]. Much later, \textit{J. D. Buckholtz} and \textit{J. K. Shaw} have shown that this condition is indeed necessary also [J. Math. Anal. Appl. 47, 626-632 (1974; Zbl 0294.30014)]. The scheme \(10\choose 10\) is the simplest scheme satisfying the Pólya condition. In this article, results analogous to those of Wider and Buckholtz-Schaw are proved for series representations derived from an arbitrary incidence scheme satisfying Pólya's condition. The value \(\pi\) is replaced by the first eigenvalue of the Pólya operator associated with the incidence scheme. The results include and extend theorems of \textit{D. Leeming} and \textit{A. Sharma} on modified Abel series expansions [Inequalities III, Proc. 3rd Symp., Los Angeles 1969, 177-199 (1972; Zbl 0302.30029)].
scientific article; zbMATH DE number 4073661
scientific article