Greedy Bases Are Best for m-Term Approximation

We study the approximation of a subset K in a Banach space X by choosing first basis B and then using n-term approximation. Into the competition for best bases we enter all unconditional bases for X. We show that if the subset K⊂X is well aligned with respect to a greedy basis B, then, in a certain sense, this basis is the best for this type of approximation. Our result improves the recent result of DeVore, Petrova and Temlyakov.