A note on best one-sided approximations

In this note we consider the relationship between best approximations and best one-sided approximations for three different measures of goodness of fit. For these measures simple relationships exist between best approximations and best one-sided approximations. In particular it is shown that a best approximation and best one-sided approximation differ only by a multiplicative constant when the measure is the uniform norm of the relative error. In this case problems involving best one-sided approximations can be reduced to problems involving best approximations. The result is especially significant if one wants to numerically determine a best one-sided approximation, since algorithms exist for numerically determining best approximations when the measure is the uniform norm of the relative error (see, for example, [1]).