Building highly conditional almost greedy and quasi-greedy bases in Banach spaces

On the Existence of Almost Greedy Bases in Banach Spaces

We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of almost greedy bases begun in [3]. We show that almost greedy bases are essentially optimal for n-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space X h...

NON-EQUIVALENT GREEDY AND ALMOST GREEDY BASES IN `p

For 1 < p < ∞ and p 6= 2 we construct a family of mutually non-equivalent greedy bases in `p having the cardinality of the continuum. In fact, no basis from this family is equivalent to a rearranged subsequence of any other basis thereof. We are able to extend this statement to the spaces Lp and H1. Moreover, the technique used in the proof adapts to the setting of almost greedy bases where sim...

Greedy Algorithms for Reduced Bases in Banach Spaces∗

Given a Banach space X and one of its compact sets F , we consider the problem of finding a good n dimensional space Xn ⊂ X which can be used to approximate the elements of F . The best possible error we can achieve for such an approximation is given by the Kolmogorov width dn(F)X . However, finding the space which gives this performance is typically numerically intractable. Recently, a new gre...

Greedy expansions in Banach spaces

Comments on the Paper ’on the Existence of Almost Greedy Bases in Banach Spaces’ By

Greedy algorithms are widely used in image processing and other applications. Let X be a real Banach space with a semi-normalized basis (en). An algorithm for n-term approximation produces a sequence of maps Fn : X → X such that, for each x ∈ X, Fn(x) is a linear combination of at most n of the basis elements (ej). Konyagin and Temlyakov [12] introduced the Thresholding Greedy Algorithm (TGA) (...