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Polynomial two-point expansions. (English) Zbl 0657.41018

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 \(f^{(n)}(0)=o(\pi^ n)\) [D. V. Widder, Trans. Am. Soc. 51, 387-398 (1942; Zbl 0027.39202)]. Much later, J. D. Buckholtz and 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 D. Leeming and A. Sharma on modified Abel series expansions [Inequalities III, Proc. 3rd Symp., Los Angeles 1969, 177-199 (1972; Zbl 0302.30029)].
Reviewer: A.Clausing

MSC:

41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
41A10 Approximation by polynomials
40A05 Convergence and divergence of series and sequences
26A51 Convexity of real functions in one variable, generalizations
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References:

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