  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.
