Lorentz spaces, approximation spaces and the greedy algorithm

In nonlinear approximation theory, understanding the structure of approximation spaces and their interplay with greedy algorithms has been a central pursuit. In this paper, we study a generalization of the classical approximation spaces associated with a wide class of bases in separable, infinite-dimensional quasi-Banach and Banach spaces, including almost greedy bases. These approximation spaces, which quantify the decay of best $n$-term approximation errors relative to a basis, are deeply connected to structural properties of the basis and the efficiency of greedy selection procedures. Using weighted Lorentz sequence spaces as a tool, we provide a comprehensive characterization of these generalized approximation spaces in terms of embeddings into and from appropriate weighted Lorentz spaces. Our results extend and unify earlier findings for greedy and unconditional bases by identifying necessary and sufficient conditions under which the classical approximation spaces, greedy approximation classes defined via the Thresholding Greedy Algorithm, and Chebyshev–greedy classes coincide. In doing so, we relax several classical assumptions—such as unconditionality and democracy—replacing them with broader notions like truncation quasi-greediness, and extend the characterizations to encompass a larger family of weights. These embeddings and equivalences elucidate the intricate relationships between approximation error decay, greedy algorithm behavior, and the fine summability captured by Lorentz norms, offering new insights into approximation theory and its connections with nonlinear analysis.