Lebesgue-type inequalities in greedy approximation

On Lebesgue-type inequalities for greedy approximation

We study the efficiency of greedy algorithms with regard to redundant dictionaries in Hilbert spaces. We obtain upper estimates for the errors of the Pure GreedyAlgorithm and the Orthogonal GreedyAlgorithm in terms of the best m-term approximations. We call such estimates the Lebesgue-type inequalities. We prove the Lebesgue-type inequalities for dictionaries with special structure. We assume t...

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...

Lebesgue-type Inequalities for Quasi-greedy Bases

Lebesgue constants for the weak greedy algorithm

We estimate the Lebesgue constants for the weak thresholding greedy algorithm in a Banach space relative to a biorthogonal system. The estimates involve the weakness (relaxation) parameter of the algorithm, as well as properties of the basis, such as its quasi-greedy constant and democracy function.

New Jensen and Ostrowski Type Inequalities for General Lebesgue Integral with Applications

Some new inequalities related to Jensen and Ostrowski inequalities for general Lebesgue integral are obtained. Applications for $f$-divergence measure are provided as well.