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

Abstract:

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 [Uniqueness of optimal piecewise polynomial $L_{1}$ approximations for generalized convex functions, from “Functional Analysis and Approximation”, Internat. Ser. Numer. Math., vol. 60 (1981), 421–432]; 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 [J. Chow, On the uniqueness of best $L_{2}[0,1]$ approximation by piecewise polynomials with variable breakpoints, Math. Comp. 39 (1982), 571–585.]. 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 [Optimal segmented polynomial $L^{p}$-approximation, Computing 26 (1981), 239–246.]. In addition, an asymptotic uniqueness result and a nonuniqueness result are indicated, which analogize known results in the $L^{2}$ case.