Lebesgue constant for Lagrange interpolation on equidistant nodes

Properties of Lebesgue function for Lagrange interpolation on equidistant nodes are investigated. It is proved that Lebesgue function can be formulated both in terms of a hypergeometric function 2F1 and Jacobt polynomials. Moreover, an integral expression of Lebesgue function is also obtained and the asymptotic behavior of Lebesgue constant is studied.     

Taylor's theorem for a function of several variablesClassroom notes

In this note the measure problem for the Lebesgue measure is discussed in terms of metric space theory. It is illuminated that under the axiom of choice most of the subsets of [0, 1) with positive outer measure are non‐Lebesgue measurable. This fact is adequate to emphasize the significance of Lebesgue measurability as well as the essentiality of the axiom of choice.     

Geometry of expanding absolutely continuous invariant measures and the liftability problem

We show that for a large class of maps on manifolds of arbitrary finite dimension, the existence of a Gibbs–Markov–Young structure (with Lebesgue as the reference measure) is a necessary as well as sufficient condition for the existence of an invariant probability measure which is absolutely continuous measure (with respect to Lebesgue) and for which all Lyapunov exponents are positive.     

Semimartingale: Itô or not ?

Itô semimartingales are the semimartingales whose characteristics are absolutely continuous with respect to Lebesgue measure. We study the importance of this assumption for statistical inference on a discretely sampled semimartingale in terms of the identifiability of its characteristics, their estimation, and propose tests of the Itô property against the non-Itô alternative when the observed semimartingale is continuous, or discontinuous with finite activity jumps, and under a number of technical assumptions.     

Analog of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> with a subadditive measure

For a subadditive fuzzy measure (not assumed finite), a Minkowski type triangle inequality, with Choquet integrals in place of Lebesgue integrals, is shown to hold. It is immediate that the set of functions for which a certain positive power of the absolute values have finite Choquet integrals is closed under addition, leading to a linear space analogous to the Lebesgue space L ^p , with a metric related to the integral of that power. Under the additional condition that the subadditive fuzzy measure is inner continuous (Sugeno), the space is shown to be complete. Consequences of the Minkowski type inequality are illustrated in two specific instances.      

Simple proof of a theorem on removable singularities of analytic functions satisfying a Lipschitz condition

Let E be a compact subset of the complex plane ?, having positive planar Lebesgue measure. Then there exists a nonconstant function f, analytic in the domain ? É, satisfying the Lipschitz condition In this note there is given a simple proof of the theorem of N. X. Uy, formulated above. It is also proved that each bounded measurable function α, defined on the set E, can be revised on a set of small Lebesgue measure so that for the function ? obtained the Cauchy integral satisfies condition (1).     

On Finite Blaschke Products whose Restrictions to the unit circle are Exact Endomorphisms

An easily checked sufficient condition is given for the restrictionof a finite Blaschke product to the unit circle to be an exactendomorphism. A formula for the entropy of such restrictionswith respect to the unique finite invariant measure equivalentto Lebesgue measure is given and it is shown that if such arestriction has maximal entropy then it is conformally equivalentto the product of a rotation and a power.     

Existence of densities for the 3D Navier–Stokes equations driven by Gaussian noise

We prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic Navier–Stokes equations in dimension $3$ . In particular, under very mild assumptions on the noise, we prove that finite dimensional projections of the solutions have densities with respect to the Lebesgue measure which have some smoothness when measured in a Besov space. This is proved thanks to a new argument inspired by an idea introduced in (Fournier and Printems. Bernoulli 16(2):343–360, 2010).     

Analytic properties of Rényi’s invariant density

Under certain conditions a many-to-one transformation of the unit interval onto itself possesses a finite invariant ergodic measure equivalent to Lebesgue measure. The purpose of this paper is to investigate these conditions and to show how differentiable and analytic properties of the invariant density are inherited from the original transformation.     

T∞-测度分解定理的进一步讨论

在对T∞-测度做进一步研究的基础上,得到了(有限或无限)T∞-测度的Hahn分解定理和Jordan分解定理。同时,用一种新方法证明了有限T∞-测度的Lebesgue分解定理。此外,还得到了一些类似于经典测度的结论。     

A finite compensation procedure for a class of two-dimensional random walks

Motivated by queueing applications, we consider a class of two-dimensional random walks, the invariant measure of which can be written as a linear combination of a finite number of product-form terms. In this work, we investigate under which conditions such an elegant solution can be derived by applying a finite compensation procedure. The conditions are formulated in terms of relations among the transition probabilities in the inner area, the boundaries as well as the origin. A discussion on the importance of these conditions is also given.     

Schrödinger operators on periodic discrete graphs

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. The spectrum of the Schrödinger operator consists of an absolutely continuous part (a union of a finite number of non-degenerated bands) plus a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. We obtain estimates of the Lebesgue measure of the spectrum in terms of geometric parameters of the graph and show that they become identities for some class of graphs. Moreover, we obtain stability estimates and show the existence and positions of large number of flat bands for specific graphs. The proof is based on the Floquet theory and the precise representation of fiber Schrödinger operators, constructed in the paper.     

On support properties of functions and their Jacobi transform

The qualitative uncertainty principle proved by Benedicks asserts that f and its Fourier transform \(\hat f\) cannot both concentrated in subsets of finite Lebesgue measure. In this paper we obtain some uncertainty principles concerning sets of finite measure in the Jacobi setting.     

On the Discreteness of the Spectrum of the magnetic Schr?dinger operator

We establish sufficient conditions for the discreteness of the spectrum of the magnetic Schr?dinger operator in terms of the Lebesgue measure.     

Application of the internal time operators for the Renyi map

Recently, the internal time operator for the Renyi map has been constructed (I. Antoniou, Z. Suchanecki, Chaos, Solitons and Fractals). It corresponds to a phase space given by the interval [0,1] and to the invariant Lebesgue measure. In this paper, following the idea of (I. Antoniou, Z. Suchanecki, Chaos, Solitons and Fractals), we construct the time operator for a dynamical system with an arbitrary invariant measure μ and an arbitrary phase space X=[a,b] with a and b finite or infinite. We illustrate also the action of such an operator on a fixed initial state.     

The Lebesgue decomposition of the free additive convolution of two probability distributions

We prove that the free additive convolution of two Borel probability measures supported on the real line can have a component that is singular continuous with respect to the Lebesgue measure on ${\mathbb{R}}$ only if one of the two measures is a point mass. The density of the absolutely continuous part with respect to the Lebesgue measure is shown to be analytic wherever positive and finite. The atoms of the free additive convolution of Borel probability measures on the real line have been described by Bercovici and Voiculescu in a previous paper.     

A variational mcshane integral characterization of the Radon-Nikodym property

We present a new characterization of Banach spaces possessing the Radon-Nikodym property in terms of additive interval functions whose McShane variational measures are absolutely continuous with respect to the Lebesgue measure.     

Dimension-Independent Estimates on the Densities of Wiener Functionals via the Log-Sobolev Inequality

Using the log-Sobolev inequality, we shall present in this note some estimates on the density of finite dimensional non-degenerate Wiener functionals which are independent on the dimension. We shall take the Gaussian measure as the reference measure, contrary to the customary choice of Lebesgue measure in the literature. As an application, we show that the limit in probability of a uniformly bounded sequence of non-degenerate Wiener functionals has a density with respect to the Gaussian measure.     

Invariant measures exist under a summability condition for unimodal maps

Summary For unimodal maps with negative Schwarzian derivative a sufficient condition for the existence of an invariant measure, absolutely continuous with respect to Lebesgue measure, is given. Namely the derivatives of the iterations of the map in the (unique) critical value must be so large that the sum of (some root of) the inverses is finite.Oblatum 7-V-1990 & 19-XI-1990Partially supported by the NWO grant.     

On the Relationship between the Dimension of the Lebesgue–Stieltjes Measure and the Rate of Approximation of a Function by Step Functions

The relationship between the rate of approximation of a monotone function by step functions (with an increasing number of values) and the Hausdorff dimension of the corresponding Lebesgue–Stieltjes measure is studied. An upper bound on the dimension is found in terms of the approximation rate, and it is shown that a lower bound cannot be constructed in these terms.