Abstract :
We consider the nonlinear approximating family Rnm of rational expressions over a real interval. In the Lp norms, 1 < p < ∞ non-normal elements of this family cannot arise as best approximations to functions outside the family. In the L1 case, Dunham (1971) has shown that for a continuous function no rational of defect two or greater, excepting the rather special case of the function 0, can be a best approximation. Cheney and Goldstein have shown (1967) that any normal rational function can arise as the best approximation to some function f ∈ L2 which is not in the rational family. We show here that there exist continuous functions not in Rnm, which do have any given defect one functions as their best approximations by using variational techniques from Wolfe (1976).
