On nonuniqueness in nonlinear L2-approximation

Abstract

It is shown that, under weak assumptions, nonlinear L2-approximation problems generally have unbounded numbers of local best approximations. This includes the rational and the exponential families of approximating functions. In addition, for a certain class of approximating families, we construct functions with three global best approximations. The results apply, for instance, to exponential and rational approximating families with one nonlinear parameter. Finally, we extend results of Spiess and Braess on the finiteness of the number of local best approximations by rational functions. © 1987.