On the Duality of Grand Bochner–Lebesgue Spaces

The associate space of the grand Bochner–Lebesgue space L^p)(I; X) is obtained without assuming the Radon–Nikodým property.

     

Lebesgue measure of the range of additive Lévy processes

We prove a theorem on the Lebesgue measure of the range of additive Lévy Processes and then use this theorem to remove Condition (1.3) of Theorem 1.5 of Khoshnevisan et al. (Ann Probab 31:1097–1141, 2003).     

Inequalities for the expectation of Δ-monotone functions

Summary For some subsets of the set of all -monotone functions on [0,1] ⁿ we characterize distribution functions F, G such that E _F fE_G f for all f within these subsets. Furthermore, we determine sharp upper and lower bounds of integrals of functions in these subsets w.r.t. all distributions with fixed marginals and give some applications of these results.     

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.     

On the Bose-Barlotti Δ-planes

Connections with flocks of quadratic cones in PG(3,q) to translation planes are used to produce Bose-Barlotti -planes that contain unitals and are of Lenz-Barlotti class II-1 (exactly one incident point-line transitivity).Dedicato ad Adriano Barlotti in occasione del suo settantesimo compleannoPartially supported by M.U.R.S.T.     

Asymptotic behavior of the evolution operator kernel expansion in powers of the time interval Δt

The upper bound for the asymptotic behavior of the coefficients of expansion of the evolution operator kernel in powers of the time intervalt was obtained. It was found that for nonpolynomial potentials, the coefficients may increase as n!. But this increase may be slower if contributions with opposite signs cancel each other. In particular, an example of the potential for which the expansion converges is presented. For polynomial potentials, thet-expansion is clearly the asymptotic one. In this case, the coefficients increase as(n L–2/L+2), where L is the order of the polynomial.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 246–255, November, 1995.     

Lebesgue points via the Poincar inequality

We show that in a Q doubling space(X, d, μ), Q 1, which satisfies a chain condition, if we have a Q Poincar′e inequality for a pair of functions(u, g) where g ∈ LQ(X), then u has Lebesgue points Hh a.e. for h(t) = log1-Q-ε(1/t). We also discuss how the existence of Lebesgue points follows for u ∈ W 1,Q(X) where(X, d, μ) is a complete Q doubling space supporting a Q Poincar′e inequality for Q 1.     

Equitable Δ-coloring of graphs

Consider a graph $G$ consisting of a vertex set $V\left(G\right)$ and an edge set $E\left(G\right)$. Let $\Delta \left(G\right)$ and $\chi \left(G\right)$ denote the maximum degree and the chromatic number of $G$, respectively. We say that $G$ is equitably $\Delta \left(G\right)$-colorable if there exists a proper $\Delta \left(G\right)$-coloring of $G$ such that the sizes of any two color classes differ by at most one. Obviously, if $G$ is equitably $\Delta \left(G\right)$-colorable, then $\Delta \left(G\right)\ge \chi \left(G\right)$. Conversely, even if $G$ satisfies $\Delta \left(G\right)\ge \chi \left(G\right)$, we cannot guarantee that $G$ must be equitably $\Delta \left(G\right)$-colorable. In 1994, the Equitable $\Delta$-Coloring Conjecture $\left(E\Delta \mathrm{CC}\right)$ asserts that a connected graph $G$ with $\Delta \left(G\right)\ge \chi \left(G\right)$ is equitably $\Delta \left(G\right)$-colorable if $G$ is different from $K_{2n+1,2n+1}$ for all $n\ge 1$. In this paper, we give necessary conditions for a graph $G$ (not necessarily connected) with $\Delta \left(G\right)\ge \chi \left(G\right)$ to be equitably $\Delta \left(G\right)$-colorable and prove that those necessary conditions are also sufficient conditions when $G$ is a bipartite graph, or $G$ satisfies $\Delta \left(G\right)\ge \frac{\left|V\left(G\right)\right|}{3}+1$, or $G$ satisfies $\Delta \left(G\right)\le 3$.     

On the Lebesgue constant of Berrut’s rational interpolant at equidistant nodes

It is well known that polynomial interpolation at equidistant nodes can give bad approximation results and that rational interpolation is a promising alternative in this setting. In this paper we confirm this observation by proving that the Lebesgue constant of Berrut’s rational interpolant grows only logarithmically in the number of interpolation nodes. Moreover, the numerical results suggest that the Lebesgue constant behaves similarly for interpolation at Chebyshev as well as logarithmically distributed nodes.     

On the Kluvánek construction of the Lebesgue integral with respect to a vector measure

The Kluvánek construction of the Lebesgue integral is extended in two directions. First, instead of a compact interval [a, b] in the real line an abstract non-empty set X is considered, instead of the ring generated by subintervals of [a, b] an arbitrary ring A of subsets of X. Secondly, instead of the length of intervals (λ([c, d]) = d?c) any vector measure λ: A→V is considered, where V is a Riesz space.     

Riemann–Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue Spaces

The present paper deals with a solution of the Riemann–Hilbert problem with piecewise continuous coefficients in the class of Cauchy type integrals with densities in grand Lebesgue spaces. Necessary and sufficient solvability condition is established. In solvability case the solutions in explicit form are constructed.     

On Value Distribution of Δ~nf

In this paper, we mainly study zeros and poles of the forward differences Δnf(z), where f(z) is a finite order meromorphic function with two Borel exceptional values.     

Density results in the Δ20 e-degrees

We show that the Δ⁰ ₂ enumeration degrees are dense. We also show that for every nonzero n-c. e. e-degree a, with n≥ 3, one can always find a nonzero 3-c. e. e-degree b such that b < a on the other hand there is a nonzero ωc. e. e-degree which bounds no nonzero n-c. e. e-degree. Received: 13 June 2000 / Published online: 3 October 2001     

Corrections to the paper “The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces”

In this paper the author proved the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with variable exponent. As an application he proved the boundedness of certain sublinear operators on the weighted variable Lebesgue space. The proof of the boundedness of the multidimensional Hardy type operator in weighted Lebesgue spaces with a variable exponent does not contain any mistakes. But in the proof of the boundedness of certain sublinear operators on the weighted variable Lebesgue space Georgian colleagues discovered a small but significant error in my paper, which was published as R.A.Bandaliev, The boundedness of certain sublinear operator in the weighted variable Lebesgue spaces, Czech. Math. J. 60 (2010), 327–337.     

On the absolute Cesàro summability of Fourier series of functions of Lebesgue classL v and some related problems in the theory of Fourier constants

Summary The author studies certain aspects of a problem on Fourier constants which emerges out of his attempt at proving a recent result ofTsuchikura on the absolute Cesàro summability of Fourier series by means of a technique that exploits properties of Fourier costants. He proves,inter alia, the Fourier-power series analogue of aconjecture on Fourier constants, which he has raised in this paper.     

Approximation of integrable functions by general linear operators of their Fourier series at the Lebesgue points

Pointwise estimates of the deviation T _n,A,B f(⋅)−f(⋅) in terms of moduli of continuity [`(w)]_·f\bar{w}_{\cdot}f and w _⋅ f are proved. Analog results on norm approximation with remarks and corollaries are also given. In the results essentially weaker conditions than those in [2, Theorem 1, p. 437] are used.     

On Hardy-Norm of Operators with Property Δ

In this paper we give sufficient conditions for which boundedness property with respect to Hardy and BMO-norms of operators Tn with property are inherited by the operator