$$L^{p}$$-Norms a nd Mahler’s Measure of Poly nomials o n the n-Dime nsio nal Torus

We prove Nikol’skii type inequalities that, for polynomials on the n-dimensional torus \(\mathbb {T}^n\), relate the \(L^p\)-norm with the \(L^q\)-norm (with respect to the normalized Lebesgue measure and \(0 <p <q < \infty \)). Among other things, we show that \(C=\sqrt{q/p}\) is the best constant such that \(\Vert P\Vert _{L^q}\le C^{\text {deg}(P)} \Vert P\Vert _{L^p}\) for all homogeneous polynomials P on \(\mathbb {T}^n\). We also prove an exact inequality between the \(L^p\)-norm of a polynomial P on \(\mathbb {T}^n\) and its Mahler measure M(P), which is the geometric mean of |P| with respect to the normalized Lebesgue measure on \(\mathbb {T}^n\). Using extrapolation, we transfer this estimate into a Khintchine–Kahane type inequality, which, for polynomials on \(\mathbb {T}^n\), relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.     

Approximation by Kantorovich-Szász Type Operators Based on Brenke Type Polynomials

In this article, we give a generalization of the Kantorovich-Szász type operators defined by means of the Brenke type polynomials introduced in the literature and obtain convergence properties of these operators by using Korovkin’s theorem. Some graphical examples using the Maple program for this approximation are given. We also establish the order of convergence by using modulus of smoothness and Peetre’s K-functional and give a Voronoskaja type theorem. In addition, we deal with the convergence of these operators in a weighted space.     

An inverse diffusion problem with nonlocal boundary conditions

This article considers the inverse problem of identification of a time‐dependent thermal diffusivity together with the temperature in an one‐dimensional heat equation with nonlocal boundary and integral overdetermination conditions when a heat exchange takes place across boundary of the material. The well‐posedness of the problem is studied under some regularity, and consistency conditions on the data of the problem together with the nonnegativity condition on the Fourier coefficients of the initial data and source term. The inverse problem is also studied numerically by using the Crank–Nicolson finite difference scheme combined with predictor‐corrector technique. The numerical examples are presented and discussed. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 564–590, 2016     

A theorem of Piccard’s type in abelian Polish groups

In the paper we prove a theorem of Piccard’s type which generalizes [9, Theorem 2]. More precisely, we show that in an abelian Polish group X the set \(\left\{ {\left( {{x_{1, \ldots ,\;}}{x_N}} \right) \in \;{X^N}\;:\;A\; \cap \;\bigcap\limits_{i = 1}^N {\left( {A + {x_i}} \right)} \;is\;not\;Haar\;meager\;in\;X} \right\}\) is a neighbourhood of 0 for every N ∈ N and every Borel non-Haar meager set A ? X. The paper refers to the paper [3].     

A measure of dependence for stable distributions

A distance based measure of dependence is proposed for stable distributions that completely characterizes independence for a bivariate stable distribution. Properties of this measure are analyzed, and contrasted with the covariation and co-difference. A sample analog of the measure is defined and demonstrated on simulated and real data, including time series and distributions in the domain of attraction of a stable law.