Let Hp denote the Lebesgue space Lp for p > 1 and the Hardy space Hp for p ≤ 1. For 0 < p, q, r < ∞, we study Hp × Hq → Hr mapping properties of bilinear operators given by finite sums of products of Calderón–Zygmund operators on stratified homogeneous Lie groups. When r ≤ 1, we show that such mapping properties hold when a number of moments of the operator vanish. This hypothesis is natural an...

Abstract In this paper we study boundedness and compactness characterizations of the commutators Cauchy type integrals on bounded strongly pseudoconvex domains D in $\mathbb C^{n}$ with boundaries $bD$ satisfying minimum regularity condition $C^{2}$ , based recent results Lanzani–Stein Duong et al. We point out that setting integral is sum essential part which a Calderón–Zygmund operator remain...

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...

We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA), a generalization of the Weak Orthogonal Matching Pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain cond...

Usually, the dynamics of linear time-invariant systems described by an integral operator of convolution type, which is defined in the Hilbert space of Lebesgue square integrable functions on the whole line. Such a description leads to contradictions. It is shown that the transition to the Hilbert space of almost periodic functions leads to the elimination of the detected inconsistencies. Multip...

Let V be a finite dimensional real vector space. Let V alsm(V ) be the space of translation invariant smooth valuations on convex compact subsets of V . Let Dens(V ) be the space of Lebesgue measures on V . The goal of the article is to construct and study an isomorphism FV : V alsm(V )−̃→V alsm(V ∗)⊗Dens(V ) such that FV commutes with the natural action of the full linear group on both spaces, ...

Motivated by the development of non-intrusive interpolation methods for parametric PDE’s in high dimension, we have introduced in [8] a sparse multi-variate polynomial interpolation procedure based on the Smolyak formula. The evaluation points lie in an infinite grid ⊗j=0Z where Z = (zj)j≥0 is any infinite sequence of mutually distinct points in some compact X in R or C. A key aspect of the int...

Abstract Let be an RD‐space, namely, a space of homogeneous type in the sense Coifman and Weiss with Borel measure μ satisfying reverse doubling condition on . Based this space, authors define multilinear strongly singular Calderón–Zygmund operator whose kernel does not need any size has more singularities near diagonal than that standard operator. For such operator, we establish its boundednes...