In this paper we show that for dictionaries with small coherence in a Hilbert space the Orthogonal Greedy Algorithm (OGA) performs almost as well as the best m−term approximation for all signals with sparsity almost as high as the best theoretically possible threshold s = 1 2 (M−1 + 1) by proving a Lebesgue-type inequality for arbitrary signals. On the other hand, we present an example of a dic...

Abstract In this short paper, I recall the history of dealing with lack compactness a sequence in case an unbounded domain and prove vanishing Lions-type result for Lebesgue-measurable functions. This lemma generalizes some results class Orlicz–Sobolev spaces. What matters here is behavior integral, not space.

We promote the concept of object directed computability in computational geometry in order to faithfully generalise the wellestablished theory of computability for real numbers and real functions. In object directed computability, a geometric object is computable if it is the effective limit of a sequence of finitary objects of the same type as the original object, thus allowing a quantitative ...

For an arbitrary innnite-dimensional Banach space X, we construct examples of strongly-measurable X-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly diierentiable; thus, for these functions the Lebesgue Diierentiation Theorem fails rather spectacularly. We also relate the degree of nondiierentiability of the indeenite Pettis integral to the cotype of X, fr...

The aim of this paper is to define the Besov-Morrey spaces and the TriebelLizorkin-Morrey spaces and to present a decomposition of functions belonging to these spaces. Our results contain an answer to the conjecture proposed by Mazzucato.

Lebesgue integration is a fundamental concept in many mathematical theories, such as real analysis, probability and information theories. Reported higher-order-logic formalizations of the Lebesgue integral either do not include, or have a limited support for the Borel algebra, which is the canonical sigma algebra used on any metric space over which the Lebesgue integral is defined. In this repo...

• Rn has a natural measure space structure; namely, Lebesgue measure m on the Borel σalgebra. The most important property of Lebesgue measure is that it is invariant under translation. This leads to nice interactions between differentiation and integration, such as integration by parts, and it gives nice functional-analytic properties to differentiation operators: for instance, the Laplacian ∆ ...

We present new results regarding Lebesgue-type inequalities for the Weak Chebyshev Greedy Algorithm (WCGA) in uniformly smooth Banach spaces. improve earlier bounds [19] dictionaries satisfying a property introduced here. apply these to derive optimal two natural examples of sequence In particular, optimality is obtained case multivariate Haar system L p with 1 < ? 2 , under Littlewood-Paley no...

We present some resolvent estimates of elliptic differential and finite-element operators in pairs of function spaces, for which the first space in a pair is endowed with stronger norm. In this work we deal with estimates in (Lebesgue, Lebesgue), (Hölder, Lebesgue), and (Hölder, Hölder) pairs of norms. In particular, our results are useful for the stability and error analysis of semidiscrete an...