Greedy expansions in Banach spaces
A 2003 Technical Report, by V. N. Temlyakov
03:06 V. N. Temlyakov
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.