It should be noted that it is not possible that a pole *z*_{i} thereby comes
to lie
in the
interval [-1,1]: the corresponding *r* could never be a best approximation
to the continuous *f*.
In particular, no *z*_{i} can be 0.

The *existence* of an optimum is easily seen.
For that purpose, write *r* as

and consider every

The *unicity* question is more involved.
The constant function example
(and, more generally,
every polynomial of degree less or equal to *N*-*P*) shows
that there are
for which no pole can be attached to *p* [4]:
every set of numbers replacing the *w*_{k} in (1) results in
(which
appears only reasonable from an approximation point of view).
In practical computations, such a large connected set of optimal points
manifests itself in the
optimization routine going around without direction.

Unicity is warranted if the optimization yields
,
i.e., if *r*=*f*on [-1,1]. Indeed, when two analytic functions *r*_{1}, *r*_{2} agree on an interval
*I*, then *r*_{1} = *r*_{2} on every domain of analyticity containing *I*, by the
fundamental lemma on analytic continuation [13, p. 150]. Therefore
if and only if
on [-1,1].
Numerically, however, the unicity shows up only if *P* is not chosen larger than the
actual number of poles of *f*, for otherwise the condition of the problem is so good
that many combinations of *P* poles yield
,
unit-roundoff error of the machine.

The unicity question can also be narrowed to an interesting one,
to which we do not have the answer.
It is obvious from the above construction that nonvanishing of the
numerator at *z*_{i} is
a sufficient condition for *r* to have a pole there. We have checked
this condition in its equivalent form [4]

Do the conditions (5)--one for every

We want to point to another representation of (5).
Indeed,
is the leading coefficient
of the polynomial interpolating a function *g* between the *x*_{k}'s,
and this coefficient
is the divided difference
of *g* with respect to all *x*_{k}'s [2]. Condition (5) can therefore
be written as

A nice property of the suggested interpolation deserves special notice:
the approximation error
cannot increase with the number of poles, this in sharp contrast with
classical rational
interpolation. Indeed,
as a new unknown, say *z*_{P}, is
added to the set of variables,
,
the optimal
value of the latter is a
feasible vector for the higher dimensional optimization--simply set
in (4).
In particular, attaching poles to the
interpolating polynomial can never worsen the quality of the approximation.

2000-05-30