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 [14],
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 [45], 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: [44] is cited. See [1, 43] for more recent work. Much of the current discussion is of spline aproximations, as in [43]. In all of the papers referenced, a detailed computational procedure must be followed to determine an approximate lower or upper bound.
In [72], 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 [72], 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 |