2 research outputs found
On the rate of convergence of greedy algorithms
We prove some results on the rate of convergence of greedy algorithms, which
provide expansions. We consider both the case of Hilbert spaces and the more
general case of Banach spaces. The new ingredient of the paper is that we bound
the error of approximation by the product of both norms -- the norm of and
the -norm of . Typically, only the -norm of is used. In
particular, we establish that some greedy algorithms (Pure Greedy Algorithm
(PGA) and its generalizations) are as good as the Orthogonal Greedy Algorithm
(OGA) in this new sense of the rate of convergence, while it is known that the
PGA is much worth than the OGA in the standard sense
Greedy expansions with prescribed coefficients in Hilbert spaces for special classes of dictionaries
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 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
dictionaries; we prove that this is the case for some classes of dictionaries
related to orthonormal sequences