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...

In this article, the spectral theory is considered, we study families and their correspondence to operators on reflexive Banach spaces; assume A a well-bounded operator Lebesgue spaces then scalar type operator. The main goals are obtain characterization of in terms associated family topology dual pairing construct continuous functional calculus for space.

Abstract In this paper, the boundedness of Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate bounds p -adic fractional weighted Lebesgue spaces. We also obtain sufficient condition commutators by taking symbol function from Lipschitz Moreover, strong type estimates for and its commutator Lorentz spaces are acquired.

Resource-bounded measure is a generalization of classical Lebesgue measure that is useful in computational complexity. The central parameter of resource-bounded measure is the resource bound ∆, which is a class of functions. When ∆ is unrestricted, i.e., contains all functions with the specified domains and codomains, resource-bounded measure coincides with classical Lebesgue measure. On the ot...

The unfolding of a vector field exhibiting a degenerate homoclinic orbit of inclination-flip type is studied. The linear part of the unperturbed system possesses a resonance but the coefficient of the corresponding monomial vanishes. We show that for an open set in the parameter space, the system possesses a suspended cubic Hénon-like map. As a consequence, strange attractors with entropy close...