It is well known that, if 1 < p < ∞, then there exists at most one best L_p-approximation, when approximating from a convex set M. On the other hand, the uniqueness of best L_p-approximation does not follow a general rule, hence, this is a question mainly of interest in the theory of best approximation. Provided f is continuous, however, we may have uniqueness when approximating from some special classes of functions.

The authors have provided an affirmative answer to the problem of uniqueness of best L_p-approximation to a continuous f∈ L₁(J₀) from the set of n-convex function, n≥ 2, J₀ being an open, bounded interval.

The goal of this paper is to prove the existence and uniqueness of the best &phgr;-approximation to a continuous f ∈ L&phgr; from L_&Pgr; for every &phgr;, under the conditions of the paper. The authors prove the existence of the best &phgr; for any f ∈ L_&phgr;. They mention that splines with infinitely many knots may be a useful tool in some applications. For example, if J is an unbounded interval, then the unique best L₁-approximation to a continuous function from the set of splines with finitely many knots necessarily has a bounded support. On the other hand, the authors develop the theory to show the splines in L₁ with infinitely many knots, and with bounded support. Finally, they prove that L_&pgr;∩ L_&phgr; satisfies Property A.