Unicity in piecewise polynomial $L^{1}$-approximation via an algorithm

Our main result shows that certain generalized convex functions on a real interval possess a unique best $L^{1}$ approximation from the family of piecewise polynomial functions of fixed degree with varying knots. This result was anticipated by Kioustelidis in [11]; however the proof given there is nonconstructive and uses topological degree as the primary tool, in a fashion similar to the proof the comparable result for the $L^{2}$ case in [5]. By contrast, the proof given here proceeds by demonstrating the global convergence of an algorithm to calculate a best approximation over the domain of all possible knot vectors. The proof uses the contraction mapping theorem to simultaneously establish convergence and uniqueness. This algorithm was suggested by Kioustelidis [10]. In addition, an asymptotic uniqueness result and a nonuniqueness result are indicated, which analogize known results in the $L^{2}$ case.