Nonlinear Equations - UFRJ

[SEC. 9.5: ALPHA-THEORY AND CONDITIONING 133
For the third statement, note that ‖X 1 ‖ ≥ (1 − β). Then
d proj (X 1 , Z) ≤ ‖X 1 − Z‖
‖X 1 ‖
≤ β 1 + β 2 + · · ·
1 − β
≤
r 1β
1 − β .
9.5 Alpha-theory and conditioning
The reproducing kernel K i (X, Y) associated to a fewspace F is analytic
in X. This implies that ¯X ↦→ K i (·, X) is also an analytic map
from M to F i . Let ρ i denote its radius of convergence, with respect
to a scaling invariant metric. Then, the value of ρ i at one point X
determines the value for all X.
In general, if
is finite, then
ρ −1
i
R −1
i
= lim sup
k≥2
( ‖D k K i (·, X)‖
k!
( ‖D k K i (·, X)‖
= sup
k≥2 k!
) 1/(k−1)
) 1/(k−1)
is also finite. This will provide bounds for the higher derivatives of
K.
Through this section, we assume for convenience that M/H = P n
and F i = H di . The unitary group U(n + 1) acts transitively on
P n . Since K i = ( ∑ X i Ȳ i ) di , ρ i = ∞ for polynomials are globally
analytic.
Taking X = e 0 and then scaling, we obtain
( ‖D k K i (·, X)‖
k!
) 1
k−1
with equality for k = 2.
(
di (d i − 1) · · · (d i − k + 1)
= ‖X‖
k!
≤
d i
2 ‖X‖
) 1
k−1