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Generalized approximate weak greedy algorithms (gAWGAs) were introduced by Galatenko and Livshits as a generalization of approximate weak greedy algorithms, which, in turn, generalize weak greedy algorithm and thus pure greedy algorithm. We consider a narrower case of gAWGA in which only a sequence of absolute errors {ξn}n=1∞{\left\{{\xi }_{n}\right\}}_{n=1}^{\infty } is nonzero. In this case sufficient condition for a convergence of a gAWGA expansion to an expanded element obtained by Galatenko and Livshits can be written as ∑n=1∞ξn2

Greedy expansions with prescribed coefficients were introduced by V. N. Temlyakov in a general case of Banach spaces. In contrast to Fourier series expansions, in greedy expansions with prescribed coefficients, a sequence of coefficients {cn}n=1∞{\left\{{c}_{n}\right\}}_{n=1}^{\infty } is fixed in advance and does not depend on an expanded element. During the expansion, only expanding elements are constructed (or, more precisely, selected from a predefined set – a dictionary). For symmetric dictionaries, V. N. Temlyakov obtained conditions on a sequence of coefficients sufficient for a convergence of a greedy expansion with these coefficients to an expanded element. In case of a Hilbert space these conditions take the form ∑n=1∞cn=∞{\sum }_{n=1}^{\infty }{c}_{n}=\infty and ∑n=1∞cn2