Based on a duality between $E$-optimality for (sub-) parameters in weighted polynomial regression and a nonlinear approximation problem of Chebyshev type, in many cases the optimal approximate designs on nonnegative and nonpositive experimental regions $\lbrack a, b\rbrack$ are found to be supported by the extrema of the only equioscillating weighted polynomial over this region with leading coefficient 1. A similar result is stated for regression on symmetric regions $\lbrack -b, b\rbrack$ for certain subparameters, provided the region is "small enough," for example, $b \leq 1$. In particular, by specializing the weight function, we obtain results of Pukelsheim and Studden and of Dette.