Practical Rational Interpolation of Exact and Inexact Data Theory ...

10 1. Classical rational interpolation
that solve
fiq(xi) − p(xi) = 0 if f(xi) is finite
q(xi) = 0 if f(xi) is ∞
Now consider the polynomial
i = 0,... ,ℓ ′ + m ′ + 2δ + s + t.
(p ∗ q − q ∗ p)(x). (1.4)
Either 0 ≤ s + t ≤ ǫ. Then (1.4) has at least ℓ ′ + m ′ + 2δ + s + t + 1 distinct
zeros, but is of degree at most
max{s,t} + ℓ ′ + m ′ + 2δ ≤ ℓ ′ + m ′ + 2δ + s + t.
Or ǫ < s + t ≤ 2ǫ. In the latter case (1.4) has at least ℓ + m + 1 =
ℓ ′ + m ′ + 2δ + ǫ + 1 distinct zeros, but is of degree at most
max{s,t} + ℓ ′ + m ′ + 2δ ≤ ℓ ′ + m ′ + 2δ + ǫ.
In both cases, (1.4) is a polynomial which must vanish identically. Thus
p(x)/q(x) and p ∗ (x)/q ∗ (x) are equivalent.
Figure 1.1: Illustration of a block.
Proposition 1.3.1 is illustrated in Figure 1.1. From a sequential application
of the Proposition then follows that the block consists of a series of
squares which do not need to be connected, but which are all symmetric
around the diagonal passing through the entry (ℓ ′ ,m ′ ).
As a corollary of the proof of Proposition 1.3.1, the degree of the deficiency
polynomial sℓ,m(x) is directly related to the shape of the block [Gut89].