Abstract:
We continue to study effeciency of approximation and convergence of greedy type algorithms in uniformly
smooth Banach spaces. This talk is based on a development of
recent results in the direction of making practical algorithms out of theoretical approximation methods. The Weak Chebyshev Greedy Algorithm (WCGA) is a general approximation method that works well in
an arbitrary uniformly smooth Banach space X for any dictionary
D.
It is an inductive procedure with each step of implementation consisting of several substeps.
We describe the first substep of a particular case of the WCGA.
Let t ε (0, 1]. Then at the first substep of the
mth step we are looking for an element φ_{m} from a given
symmetric dictionary D satisfying
(1) F_{fm1}(φ_{m}) ≥
t sup F_{fm1}(g)
g ε
D
where f_{m1} is a residual after (m1)th step
and F_{fm1} is a norming functional of
f_{m1}. It is a greedy step of the WCGA. It is clear that in the case of
infinite dictionary D there is no direct computationally feasible way of evaluating
sup_{gεD}F_{fm1}(g). This is
the main issue that we address in the talk. We consider countable dictionaries
D = {±ψ_{j}}_{j=1 to ∞} and replace the
inequality used above by
F_{fm1}(φ_{m}) ≥
t sup  F_{fm1}(ψ_{j}),
φ_{m}
ε
{±ψ_{j}}_{j=1 to Nm}
1 ≤ j ≤ N_{m}
The
restriction j ≤ N_{m} is known in the literature as the depth search condition. We prove convergence and rate of
convergence results for such a modification of the WCGA.
^{1}This research was supported by the National Science
Foundation Grant DMS 0200187
[LECTURE SLIDES]
