AbstractFor trigonometric polynomial approximation on a circle, the century-old de la
Vallée-Poussin construction has attractive features: it exhibits uniform convergence for
all continuous functions as the degree of the trigonometric polynomial goes to infinity,
yet it also has arbitrarily fast convergence for sufficiently smooth functions. This paper
presents an explicit generalization of the de la Vallée-Poussin construction to higher
dimensional spheres S^d ≤ R^{d+1}. The generalization replaces the C^∞ filter introduced by
Rustamov by a piecewise polynomial of minimal degree. For the case of the circle the filter
is piecewise linear, and recovers the de la Vallée-Poussin construction, while for the
general sphere S^d the filter is a piecewise polynomial of degree d and smoothness C^{d−1}.
In all cases the approximation converges uniformly for all continuous functions, and has
arbitrarily fast convergence for smooth functions.
Polynomial approximation on spheres - generalizing de la Vallée-Poussin