A factorization of the Askey's characteristic function \((1-\| t\| _{2n+1})_ +^{n+1}\)
In 1973 \textit{R. Askey} [Radial characteristic functions. Math. Res. Center, Univ. Wisconsin, Madison. Tech. Rep. No.1262 (1973)] proved the following generalization of Polya's theorem (corresponding to the case \(n=0):\) Let g: [0,\(\infty)\to R\) be continuous such that \(g(0)=1\), \((- 1)^ ng^{(n)}\) exists and is convex in (0,\(\infty)\) and g(t)\(\to 0\), \(g^{(n)}(t)\to 0\) as \(t\to \infty\). Then \(g\circ \| \cdot \|\) is a characteristic function on \(R^{2n+1}\), i.e. is the Fourier transform of a probability distribution on \(R^{2n+1}.\) In the paper under review a new proof of Askey's theorem is offered. The technique of the proof is to locate the zeros of the polynomial \(\int^{1}_{z}(1-t^ 2)^ ndt\).
scientific article; zbMATH DE number 3969758