From ultraspherical to Hermite polynomial expansions via Poincaré's central limit theorem (Q1596344)

Expansions in terms of ultraspherical polynomials \(C_n^\lambda (x)\) have as limiting cases Fourier series \((\lambda\to 0)\) and power series \((\lambda\to\infty)\); this was discussed recently by \textit{R. Askey} and \textit{P. T. Haimo} [Am. Math. Monthly 103, No. 4, 297-304 (1996; Zbl 0854.33005)]. In the present paper a different approach is seen. First, the polynomials are introduced by consideration of the unit sphere in \(\mathbb{R}^{D+1}\), rather than the interval \([-1;1]\). Next, instead of being fixed, the radius of the sphere is allowed to be proportional to \(\sqrt D\). The expansion now behaves differently in the upper limit; in fact an expansion in terms of Hermite polynomials is obtained. Questions of convergence, especially the conditions on the function to be expanded, are discussed in detail.