Greedy expansions in Banach spaces
A 2003 Technical Report, by V. N. Temlyakov
03:06 V. N. Temlyakov
Greedy expansions in Banach spaces
We study convergence and rate of convergence of expansions of elements in a Banach space X into series with regard to a given dictionary D. For convenience we assume that D is symmetric: implies
. The primary goal of this paper is to study representations of an element
by a series
In building such a representation we should construct two sequences: and
. In this paper the construction of
will be based on ideas used in greedy-type nonlinear approximation. This explains the use of the term greedy expansion. We use a norming function
of a residual
obtained after
steps of an expansion prodedure to select the mth element
from the dictionary. This approach has been used in previous papers on greedy approximation. The new feature of this paper is a way of selecting the mth coefficient
of the expansion. An approach developed in the paper works in any uniformly smooth Banach space. For instance, in a Banach space X with the modulus of smoothness
we can choose
from the equation
where is the weakness parameter of an algorithm and
is its tuning parameter. We prove convergence of such expansions for all
and obtain rate of convergence for
- the closure (in X) of the convex hull of D.