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 *n*th 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 |