Bidemocratic Bases and Their Connections with Other Greedy-Type Bases

Lebesgue-type Inequalities for Quasi-greedy Bases

We show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C logN . We show with two examples that this bound is a...

Renorming spaces with greedy bases

In approximation theory one is often faced with the following problem. We start with a signal, i.e., a vector x in some Banach space X. We then consider the (unique) expansion ∑∞ i=1 xiei of x with respect to some (Schauder) basis (ei) of X. For example, this may be a Fourier expansion of x, or it may be a wavelet expansion in Lp. We then wish to approximate x by considering m-term approximatio...

Lebesgue-type Inequalities for Quasi-greedy Bases

Greedy approximation with regard to non-greedy bases

Peixin Ye Nankai University, China [email protected] In this talk we discuss which properties of a basis are important for certain direct and inverse theorems in nonlinear approximation. We study greedy approximation with regard to bases with different properties. We consider bases that are tensor products of univariate greedy bases. Some results known for unconditional bases, such as Lebesgue...

Greedy Bases for Besov Spaces

We prove that the Banach spaces (⊕n=1`p )`q , which are isomorphic to the Besov spaces on [0, 1], have greedy bases, whenever 1 ≤ p ≤ ∞ and 1 < q < ∞. Furthermore, the Banach spaces (⊕n=1`p )`1 , with 1 < p ≤ ∞, and (⊕n=1`p )c0 , with 1 ≤ p < ∞ do not have a greedy bases. We prove as well that the space (⊕n=1`p )`q has a 1-greedy basis if and only if 1 ≤ p = q ≤ ∞.