Spherical analog of the complex form of trigonometric Fourier series; approximation (Q5942083)

First, the author discusses peculiarities of the complex form of a trigonometric Fourier series. In order to extend them to the multivariate case he suggests a spherical analog with algebraic structure similar to that of the usual Fourier components. For example, in dimension 4 and 8 this coincides with the division rings of quaternions and Cayley numbers, respectively. For the analog of partial Fourier sums obtained, the deviations of them from the function are estimated with respect to various norms; the cases of even and odd dimension are treated separately.