Higher-Order Hermite-Fejér Interpolation for Stieltjes Polynomials

Let ( ) := (1− 2 ) −1/2 and , be the ultraspherical polynomials with respect to ( ). Then, we denote the Stieltjes polynomials . In this paper, we consider the higher-order Hermite-Fejér interpolation operator +1, based on the zeros of , +1 and the higher order extended Hermite-Fejér interpolation operator H 2 +1, based on the zeros of , +1 , . When m is even, we show that Lebesgue constants of these interpolation operators are ( max{(1− ) −2,0} )(0 < < 1) and ( max{(1−2 ) −2,0} )(0 < < 1/2), respectively; that is, ‖H 2 +1, ‖ = ( max{(1−2 ) −2,0} )(0 < < 1) and ‖ +1, ‖ = ( max{(1− ) −2,0} )(0 < < 1/2). In the case of the Hermite-Fejér interpolation polynomials H 2 +1, [⋅] for 1/2 ≤ < 1, we can prove the weighted uniform convergence. In addition, when m is odd, we will show that these interpolations diverge for a certain continuous function on [−1, 1], proving that Lebesgue constants of these interpolation operators are similar or greater than log n

Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses

Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses

Fractional Calculus - Theory and Applications

In recent years, fractional calculus has led to tremendous progress in various areas of science and mathematics. New definitions of fractional derivatives and integrals have been uncovered, extending their classical definitions in various ways. Moreover, rigorous analysis of the functional properties of these new definitions has been an active area of research in mathematical analysis. Systems considering differential equations with fractional-order operators have been investigated thoroughly from analytical and numerical points of view, and potential applications have been proposed for use in sciences and in technology. The purpose of this Special Issue is to serve as a specialized forum for the dissemination of recent progress in the theory of fractional calculus and its potential applications

The early historical roots of Lee-Yang theorem

A deep and detailed historiographical analysis of a particular case study concerning the so-called Lee-Yang theorem of theoretical statistical mechanics of phase transitions, has emphasized what real historical roots underlie such a case study. To be precise, it turned out that some well-determined aspects of entire function theory have been at the primeval origins of this important formal result of statistical physics.Comment: History of Physics case study. arXiv admin note: substantial text overlap with arXiv:1106.4348, arXiv:math/0601653, arXiv:0809.3087, arXiv:1311.0596 by other author