Determining polynomial upper and lower bounds of general functions has been discussed in the literature [45, 14, 18, 72]. In the results cited, optimality is determined via the norm, as done in section .
In , it is shown that if is bounded, and finite for at least n+1 points, then there exists optimal lower and upper degree n bounds. It is also shown that if g is continuous on [0,1], and differentiable on (0,1), then the optimal bounds are unique. It is also established that for g with or , the optimal bounds are found by interpolating g and , as we have done. The optimal interpolation points are shown to be the nodes of a Gauss quadrature formula. With linear bounds, this corresponds to interpolating the value of g for and , or interpolating the value and derivative of g for .
In , a collection of constrained approximation problems is brought together, with one-sided approximation treated as a special case of general constrained approximation problems. Linear programming is suggested as a method to determine bounds when the nth derivative crosses zero:  is cited. See [1, 43] for more recent work. Much of the current discussion is of spline aproximations, as in . In all of the papers referenced, a detailed computational procedure must be followed to determine an approximate lower or upper bound.
In , another approach to proving upper and lower polynomial bounds optimal is taken. With this approach, it is shown that interpolating the value and/or derivative at the nodes of a Gauss quadrature formula constructs the optimal polynomial bound, provided that the bound does not interpolate the function elsewhere.
Most of these results generalize to non-polynomial bounds. Often, the bounds are taken from a Chebyshev system [18, 45, 1, 44]; the set of n degree polynomials form a Chebyshev system. In , bounds are taken as linear combinations of an arbitrary set of continuous functions. Characterizations of constrained approximation solutions has also been studied [30, 70].
|Jeff Tupper||March 1996|