Sherman transformations for functions on the sphere (Q1074052)

This paper is a continuation of work by \textit{T. O. Sherman} [Trans. Am. Math. Soc. 209, 1-31 (1975; Zbl 0308.43009)] and \textit{M. Morimoto} [Analytic functionals on the sphere and their Fourier-Borel transformations, Complex Analysis, Banach Cent. Publ. 11, 223-250 (1983)]. Let S denote the unit sphere in \({\mathbb{R}}^{d+1}\). Functions and functionals f on S can be expanded in terms of the classical spherical harmonics in \(d+1\) dimensions. Using this expansion, the Sherman transform associates to f an infinite sequence of polynomials on the equator of S, and that sequence has properties that reflect those of f itself. Put \(\tilde S=\{z\in {\mathbb{C}}^{d+1} :\) \(z^ 2_ 1+...+z^ 2_{d+1}=1\}\) (so that \(S=\tilde S\cap {\mathbb{R}}^{d+1})\) and Exp \(\tilde S=\) restriction of entire functions of exponential type in \({\mathbb{C}}^{d+1}\) to \(\tilde S.\) The author takes a string of function spaces between Exp \(\tilde S\) and \(L^ 2(S)\) and studies the image under the Sherman transformation of these spaces as well as that of their duals. These images involve the growth of the sequence of norms of the ''Sherman polynomials''. The author also defines a sort of dual of the Sherman transform and carries out a similar study for it. Finally, the relation of these transforms to the Fantappiè indicator is pointed out.