ABSTRACT. Let {G_{k}} be a sequence of linear subspaces of a normed linear space E. We want to characterize those sequences {g_{k}} in E, with g_{k} ∈ G_{k} (k = 1, 2, ...), for which there exists an x ∈ E such that

g_{k} ∈ P_{G_{k}}(x) (k = 1, 2, ...),

where P_{G}(x) denotes the set of all elements of best approximation to x from G, i.e., the set

{g_{0} ∈ G | ||x − g_{0}|| = inf_{g∈G} ||x − g||}.

This general study was motivated by a problem posed by T.J. Rivlin [5] who asked: Characterize (intrinsically) those n-tuples of polynomials (p_{0}, p_{1}, ..., p_{n−1}), with degree of p_{j} equal to j for all j, for which there exists an x ∈ C[0, 1] such that the polynomial of best approximation to x, from the space of polynomials P_{j} of degree at most j, Π_{P_{j}}(x), is p_{j} (j = 0, 1, ..., n − 1). What is the answer in the particular case where n = 2?

We can give a complete (but not intrinsic) characterization of our general problem (Theorem 1). This allows us, for example, to get a complete characterization of the general problem in a Hilbert space when the subspaces G_{k} are increasing (Theorem 5), or when they are decreasing (Theorem 6). In C[0, 1], we answer Rivlin’s second question as follows. Given p_{j} ∈ P_{j}(j = 0, 1), there exists x ∈ C[0, 1] such that Π_{P_{j}}(x) = p_{j} (j = 0, 1) if and only if p_{1} − p_{0} is either identically zero or changes sign once in [0, 1].