Abstract
We establish a characterization for an [Formula: see text]-manifold [Formula: see text] to admit [Formula: see text] functions [Formula: see text] and [Formula: see text] functions [Formula: see text] in [Formula: see text] so that every element of [Formula: see text] can be approximated by rational combinations of [Formula: see text] and polynomial combinations of [Formula: see text]. As an application, we show that the optimal value of [Formula: see text] and [Formula: see text] for all manifolds of dimension [Formula: see text] is [Formula: see text], when [Formula: see text] and [Formula: see text].
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