Lebesgue Type Research Articles

Planar substitutions to Lebesgue type space-filling curves and relatively dense fractal-like sets in the plane

We present an algorithm for generating curves filling the unit square; i.e. space-filling curves, from any given planar substitution satisfying a mild condition. The proposed algorithm is mimicking construction steps of Lebesgue's curve and is based on linear interpolation. Generated space-filling curves for some known substitutions are elucidated. Some of those substitutions further induce relatively dense fractal-like sets in the plane, whenever some additional assumptions are satisfied.

Lebesgue sampling approach to robust stabilization of boolean control networks with external disturbances

This paper introduces the Lebesgue sampling approach to the robust stabilization of Boolean control networks (BCNs) with external disturbances. Given a Lebesgue sampling region and a feedback control, a time aggregated system is obtained via the semi-tensor product method. Then, a new criterion is presented for the robust stabilization of time aggregated system. Furthermore, given a signal of Lebesgue sampling, a sequence of the Lebesgue type robust reachable sets is constructed. Based on these reachable sets, several algorithms are presented to design both Lebesgue sampling region and sampled-data state feedback control for the robust stabilization of BCNs.

Lebesgue type decompositions and Radon–Nikodym derivatives for pairs of bounded linear operators

For a pair of bounded linear Hilbert space operators A and Bone considers the Lebesgue type decompositions of B with respect toA into an almost dominated part and a singular part, analogous to theLebesgue decomposition for a pair of measures in which case one speaks ofan absolutely continuous and a singular part. A complete parametrizationof all Lebesgue type decompositions will be given, and the uniqueness ofsuch decompositions will be characterized. In addition, it will be shownthat the almost dominated part of B in a Lebesgue type decomposition hasan abstract Radon–Nikodym derivative with respect to the operator A.

Spectral properties of some unions of linear spaces

We consider additive spaces, consisting of two intervals of unit length or two general probability measures on R1, positioned on the axes in R2, with a natural additive measure ρ. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of L2(ρ) and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on R1. We show that non-overlapping additive spaces possess Riesz bases, and we give a necessary condition for overlapping spaces. We also show that some overlapping additive spaces of Lebesgue type have exponential orthonormal bases, while some do not. A particular example is the L shape at the origin, which has a unique orthonormal basis up to translations of the form{e2πi(λ1x1+λ2x2):(λ1,λ2)∈Λ}, whereΛ={(n/2,−n/2)|n∈Z}.

Sharp approximation theorems and Fourier inequalities in the Dunkl setting

In this paper we study direct and inverse approximation inequalities in Lp(Rd), 1<p<∞, with the Dunkl weight. We obtain these estimates in their sharp form substantially improving previous results. We also establish new estimates of the modulus of smoothness of a function f via the fractional powers of the Dunkl Laplacian of approximants of f. Moreover, we obtain new Lebesgue type estimates for moduli of smoothness in terms of Dunkl transforms. Needed Pitt-type and Kellogg-type Fourier–Dunkl inequalities are derived.

Convergence results in Birkhoff weak integrability

This article presents some properties of real Birkhoff weak integrable functions with respect to a non-additive set function, such as H??lder inequality, Minkowski inequality, Lebesgue type convergence theorems and Fatou type theorem.

Uniformly consistently estimating the proportion of false null hypotheses via Lebesgue–Stieltjes integral equations

The proportion of false null hypotheses is a very important quantity in statistical modelling and inference based on the two-component mixture model and its extensions, and in control and estimation of the false discovery rate and false non-discovery rate. Most existing estimators of this proportion threshold p-values, deconvolve the mixture model under constraints on its components, or depend heavily on the location-shift property of distributions. Hence, they usually are not consistent, applicable to non-location-shift distributions, or applicable to discrete statistics or p-values. To eliminate these shortcomings, we construct uniformly consistent estimators of the proportion as solutions to Lebesgue–Stieltjes integral equations. In particular, we provide such estimators respectively for random variables whose distributions have Riemann–Lebesgue type characteristic functions, whose distributions form discrete natural exponential families with infinite supports, and whose distributions form natural exponential families with separable moment sequences. We provide the speed of convergence and uniform consistency class for each such estimator under independence. In addition, we provide two examples for which a consistent estimator of the proportion cannot be constructed using our techniques.

Fredholm-Choquet integral equations

In this paper we study the classical Fredholm integral equation of second kind, in which the Lebesgue type integral is replaced by the more general Choquet integral with respect to a monotone, submodular and continuous from below set function, including the so-called distorted Lebesgue measures.

Lebesgue type decompositions for linear relations and Ando’s uniqueness criterion

A linear relation, i.e., a multivalued operator T from a Hilbert space h to a Hilbert space k has Lebesgue type decompositions T = T-1 + T-2, where T i is a closable operator and T-2 is an operator or relation which is singular. There is one canonical decomposition, called the Lebesgue decomposition of T, whose closable part is characterized by its maximality among all closable parts in the sense of domination. All Lebesgue type decompositions are parametrized, which also leads to necessary and sufficient conditions for the uniqueness of such decompositions. Similar results are given for weak Lebesgue type decompositions, where T-1 is just an operator without being necessarily closable. Moreover, closability is characterized in different useful ways. In the special case of range space relations the above decompositions may be applied when dealing with pairs of (nonnegative) bounded operators and nonnegative forms as well as in the classical framework of positive measures.

Generalized Lebesgue Points for Hajłasz Functions

Let X be a quasi-Banach function space over a doubling metric measure space P. Denote by αX the generalized upper Boyd index of X. We show that if αX&lt;∞ and X has absolutely continuous quasinorm, then quasievery point is a generalized Lebesgue point of a quasicontinuous Hajłasz function u∈M˙s,X. Moreover, if αX&lt;(Q+s)/Q, then quasievery point is a Lebesgue point of u. As an application we obtain Lebesgue type theorems for Lorentz–Hajłasz, Orlicz–Hajłasz, and variable exponent Hajłasz functions.

Some questions of harmonic analysis in weighted Morrey type spaces

Weighted Morrey-type classes of functions that are harmonic in the unit disk and in the upper half plane are defined in this work. Under some conditions on the weight function, we study some properties of functions belonging to these classes. Estimation of the maximum values of harmonic functions for a nontangential angle through the Hardy–Littlewood maximal function are generalized to more general case, and then the boundedness of Hardy–Littlewood operator is applied in the Morrey-type spaces. Weighted Morrey–Lebesgue type space is defined, where the shift operator is continuous with respect to shift, and its invariance with regard to the singular operator is proved. The validity of Minkowski inequality in Morrey–Lebesgue type spaces is also proved. An approximation properties of the Poisson kernel are studied.

On Morrey-type classes of harmonic functions

Weighted Morrey-type classes of functions that are harmonic in the unit disk and in the upper half plane are defined in this work. Under some conditions on the weight function, we study some properties of functions belonging to these classes. Maximum values of harmonic functions for a nontangential angle are estimated through the Hardy–Littlewood maximal function, and then the boundedness of Hardy–Littlewood operator is applied in the Morrey-type spaces. Weighted Morrey–Lebesgue type space is defined, where the shift operator is continuous with respect to shift, and its invariance with regard to the singular operator is proved. The validity of Minkowski inequality in Morrey–Lebesgue type spaces is also proved.

New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications

We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadratic-phase Fourier integral operators are also studied (including a Riemann–Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity). As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.

Integral of Non Positive Functions

Summary In this article, we formalize in the Mizar system [1, 7] the Lebesgue type integral and convergence theorems for non positive functions [8],[2]. Many theorems are based on our previous results [5], [6].

Lebesgue inequalities for the greedy algorithm in general bases

We present various estimates for the Lebesgue type inequalities associated with the thresholding greedy algorithm, in the case of general bases in Banach spaces. We show the optimality of the involved constants in some situations. Our results recover and slightly improve various estimates appearing earlier in the literature.

Grothendieck–Lidskii trace formula for mixed-norm and variable Lebesgue spaces

In this note we present the metric approximation property for weighted mixed-norm L_w^{(p_1,\dots, p_n)} and variable exponent Lebesgue type spaces. As a consequence, this also implies the same property for modulation and Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces and state the corresponding Grothendieck–Lidskii trace formulae. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on Rn and functions of the harmonic oscillator in terms of global symbols.

Inhomogeneous microlocal propagation of singularities in Fourier Lebesgue spaces

In the paper some results of microlocal continuity for pseudodifferential operators whose symbols belong to weighted Fourier Lebesgue spaces are given. Inhomogeneous local and microlocal propagation of singularities of Fourier Lebesgue type are then studied, with applications to some classes of semilinear equations.

On the mappings connected with parallel addition of nonnegative operators

We study a mapping \(\tau _G\) of the cone \({\mathbf B}^+({\mathcal H})\) of bounded nonnegative self-adjoint operators in a complex Hilbert space \({\mathcal H}\) into itself. This mapping is defined as a strong limit of iterates of the mapping \({\mathbf B}^+({\mathcal H})\ni X\mapsto \mu _G(X)=X-X:G\in {\mathbf B}^+({\mathcal H})\), where \(G\in {\mathbf B}^+({\mathcal H})\) and X : G is the parallel sum. We find explicit expressions for \(\tau _G\) and describe its properties. In particular, it is shown that \(\tau _G\) is sub-additive, homogeneous of degree one, and its image coincides with the set of its fixed points which is a subset of \({\mathbf B}^+({\mathcal H})\), consisting of all Y such that \(\mathrm{ran\,} Y^{\frac{1}{2}}\cap \mathrm{ran\,} G^{\frac{1}{2}}=\{0\}\). Relationships between \(\tau _G\) and Lebesgue type decomposition of nonnegative self-adjoint operator are established and applications to the properties of unbounded self-adjoint operators with trivial intersections of their domains are given.

Greedy bases in variable Lebesgue spaces

We compute the right and left democracy functions of admissible wavelet bases in variable Lebesgue spaces defined on $R^n$. As an application we give Lebesgue type inequalities for these wavelet bases. We also show that our techniques can be easily modified to prove analogous results for weighted variable Lebesgue spaces and variable exponent Triebel-Lizorkin spaces.

Lebesgue Decomposition Theorem and Weak Radon-Nikodým Theorem for Generalized Fuzzy Number Measures

The Lebesgue type decomposition theorem and weak Radon-Nikodým theorem for fuzzy valued measures in separable Banach spaces are established.