The authors generalize a result of \textit{G. Wahba} [SIAM J. Sci. Stat. Comput. 2, 5-16 (1981; Zbl 0537.65008)] concerning spline interpolation and smoothing on the two-dimensional sphere. Their generalizations are threefold. Firstly, Wahba's results are extended to arbitrary dimensional hyperspheres. Secondly, instead of powers of the Laplace-Beltrami operator, more general operators are considered. Finally, the decomposition of the underlying function spaces is changed to permit more flexible combinations of hyperspherical harmonics and hyperspherical splines for interpolation and smoothing schemes. Numerical tests are presented to illustrate the behaviour of such schemes for several test functions in different dimensions.