Greedy expansions with prescribed coefficients in Hilbert spaces for special classes of dictionaries

TL;DR

This paper investigates the convergence of greedy algorithms with fixed coefficients in Hilbert spaces, identifying conditions that can be relaxed for specific classes of spaces and dictionaries, including finite-dimensional and orthonormal-related sets.

Contribution

It demonstrates that necessary and sufficient conditions for convergence can be weakened for certain classes of spaces and dictionaries, especially finite-dimensional and orthonormal-related sets.

Findings

Convergence conditions are relaxable in finite-dimensional spaces.

Certain classes of dictionaries allow weaker convergence conditions.

The results extend the understanding of greedy algorithms in specialized Hilbert space settings.

Abstract

Greedy expansions with prescribed coefficients have been introduced by V. N. Temlyakov in the frame of Banach spaces. The idea is to choose a sequence of fixed (real) coefficients $\{c_n\}_{n=1}^\infty$ and a fixed set of elements (dictionary) of the Banach space; then, under suitable conditions on the coefficients and the dictionary, it is possible to expand all the elements of the Banach space in series that contain only the fixed coefficients and the elements of the dictionary. In Hilbert spaces the convergence of greedy algorithm with prescribed coefficients is characterized, in the sense that there are necessary and sufficient conditions on the coefficients in order that the algorithm is convergent for all the dictionaries. This paper is concerned with the question if such conditions can be weakened for particular classes of spaces or dictionaries; we prove that this is the case for finite dimensional spaces, and for some classes of dictionaries related to orthonormal sequences in infinite dimensional spaces.