scholarly journals Another Hilbert inequality and critically separated interpolation nodes

PAMM ◽  
2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Stefan Kunis ◽  
Julian Rolfes
Keyword(s):  
Hilbert Inequality ◽  
Interpolation Nodes

On a Refinement of Hardy-Hilbert Inequality

2012 ◽  
Vol 166-169 ◽  
pp. 3027-3030
Author(s):  
Bao Ju Sun
Keyword(s):  
Weight Coefficient ◽  
Hilbert Inequality ◽  
Maximum Minimum

In this paper, by using Maximum-minimum monotonic theorem and estimating the weight coefficient, a refinement of Hardy- Hilbert inequality is established.

scholarly journals Rational interpolation of the function |x|α by an extended system of Chebyshev – Markov nodes

2020 ◽  
Vol 55 (4) ◽  
pp. 391-405
Author(s):  
E. A. Rovba ◽  
V. Yu. Medvedeva
Keyword(s):  
Rational Functions ◽  
Rational Interpolation ◽  
Asymptotic Equality ◽  
Fixed Number ◽  
Theoretical Research ◽  
Extended System ◽  
Uniform Approximations ◽  
Upper Estimate ◽  
Interpolation Functions ◽  
Interpolation Nodes

In this paper, we study the approximations of a function |x|α, α > 0 by interpolation rational Lagrange functions on a segment [–1,1]. The zeros of the even Chebyshev – Markov rational functions and a point x = 0 are chosen as the interpolation nodes. An integral representation of an interpolation remainder and an upper bound for the considered uniform approximations are obtained. Based on them, a detailed study is made:a) the polynomial case. Here, the authors come to the famous asymptotic equality of M. N. Hanzburg;b) at a fixed number of geometrically different poles, the upper estimate is obtained for the corresponding uniform approximations, which improves the well-known result of K. N. Lungu;c) when approximating by general Lagrange rational interpolation functions, the estimate of uniform approximations is found and it is shown that at the ends of the segment [–1,1] it can be improved.The results can be applied in theoretical research and numerical methods. 

FOURIER FORMULAE FOR EQUIDISTANT HERMITE TRIGONOMETRIC INTERPOLATION

2011 ◽  
Vol 04 (01) ◽  
pp. 127-144 ◽  
Author(s):  
Arnak Poghosyan
Keyword(s):  
Convergence Acceleration ◽  
Lanczos Method ◽  
Trigonometric Interpolation ◽  
Quadrature Formulae ◽  
Interpolation Polynomials ◽  
Exact Constants ◽  
Interpolation Nodes

A sequence of Hermite trigonometric interpolation polynomials with equidistant interpolation nodes and uniform multiplicities is investigated. We derive relatively compact formula that gives the interpolants as functions of the coefficients in the DFTs of the derivative values. The coefficients can be calculated by the FFT algorithm. Corresponding quadrature formulae are derived and explored. Convergence acceleration based on the Krylov-Lanczos method for accelerating both the convergence of interpolation and quadrature is considered. Exact constants of the asymptotic errors are obtained and some numerical illustrations are presented.

Curve Parameterization and Curvature via Method of Hurwitz-Radon Matrices

2011 ◽  
Vol 16 (1-2) ◽  
pp. 49-56 ◽  
Author(s):  
Dariusz Jakóbczak
Keyword(s):  
Computer Vision ◽  
Image Analysis ◽  
Object Recognition ◽  
Explicit Form ◽  
Local Maximum ◽  
Parametric Representation ◽  
Feature Points ◽  
Implicit Representation ◽  
Computer Vision Applications ◽  
Interpolation Nodes

Curve Parameterization and Curvature via Method of Hurwitz-Radon MatricesParametric representation of the curve is more appropriate in computer vision applications then explicit formy=f(x)or implicit representationf(x, y) = 0. Proposed method of Hurwitz-Radon Matrices (MHR) can be used in parameterization and interpolation of curves in the plane. Suitable parameterization leads to curvature calculations. Points with local maximum curvature are treated as feature points in object recognition and image analysis. This paper contains the way of curve parameterization and computing the curvature in the range of two successive interpolation nodes via MHR method. Proposed method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The operator of Hurwitz-Radon (OHR), built from these matrices, is described. It is shown how to create the orthogonal OHR and how to use it in a process of curve parameterization and curvature calculation.

scholarly journals Approximate Solutions of Fisher's Type Equations with Variable Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Differential Equations ◽  
Legendre Polynomials ◽  
Nonlinear Partial Differential Equations ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Variable Coefficients ◽  
Time Dependent ◽  
First Order ◽  
Spatial Derivatives ◽  
Interpolation Nodes

The spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of time-dependent Fisher’s type problems. The spatial derivatives are collocated at a Legendre-Gauss-Lobatto interpolation nodes. The proposed method has the advantage of reducing the problem to a system of ordinary differential equations in time. The four-stage A-stable implicit Runge-Kutta scheme is applied to solve the resulted system of first order in time. Numerical results show that the Legendre-Gauss-Lobatto collocation method is of high accuracy and is efficient for solving the Fisher’s type equations. Also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations.

scholarly journals On Hilbert's Inequality for Double Series and Its Applications

2008 ◽  
Vol 2008 ◽  
pp. 1-12
Author(s):  
Zhou Yu ◽  
Gao Mingzhe
Keyword(s):  
Real Function ◽  
Hardy Inequality ◽  
Double Series ◽  
Cauchy Inequality ◽  
Hilbert Inequality ◽  
Hilbert’S Inequality ◽  
Riesz Inequality

This study shows that a refinement of the Hilbert inequality for double series can be established by introducing a real functionu(x)and a parameterλ. In particular, some sharp results of the classical Hilbert inequality are obtained by means of a sharpening of the Cauchy inequality. As applications, some refinements of both the Fejer-Riesz inequality and Hardy inequality inHpfunction are given.

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