Some biorthogonal polynomials arising in numerical analysis and approximation theory

2022 ◽  
Vol 403 ◽  
pp. 113842
Author(s):  
D.S. Lubinsky ◽  
A. Sidi
2014 ◽  
Vol 2014 ◽  
pp. 1-2 ◽  
Author(s):  
Sofiya Ostrovska ◽  
Elena Berdysheva ◽  
Grzegorz Nowak ◽  
Ahmet Yaşar Özban

Author(s):  
Y. A. Rouba ◽  
K. A. Smatrytski ◽  
Y. V. Dirvuk

In this paper we propose a new approach to the construction of quadrature formulas of interpolation rational type on an interval. In the introduction, a brief analysis of the results on the topic of the research is carried out. Most attention is paid to the works of mathematicians of the Belarusian school on approximation theory – Gauss, Lobatto, and Radau quadrature formulas with nodes at the zeros of the rational Chebyshev – Markov fractions. Rational fractions on the segment, generalizing the classical orthogonal Jacobi polynomials with one weight, are defined, and some of their properties are described. One of the main results of this paper consists in constructing quadrature formulas with nodes at zeros of the introduced rational fractions, calculating their coefficients in an explicit form, and estimating the remainder. This result is preceded by some auxiliary statements describing the properties of special rational functions. Classical methods of mathematical analysis, approximation theory, and the theory of functions of a complex variable are used for proof. In the conclusion a numerical analysis of the efficiency of the constructed quadrature formulas is carried out. Meanwhile, the choice of the parameters on which the nodes of the quadrature formulas depend is made in several standard ways. The obtained results can be applied for further research of rational quadrature formulas, as well as in numerical analysis.


Author(s):  
Ya. A. Rouba ◽  
K. A. Smatrytski ◽  
Ya. V. Dirvuk

In this paper, a new approach to the definition of the interpolating rational process of Fejer – Hermite with first-type Chebyshev – Markov nodes on a segment is studied and some of its approximating properties are described. In the introduction a brief analysis of the results on the topic of the research is carried out. Herein, the methods of the construction of interpolating processes, in particular, Fejer – Hermite processes, in the polynomial and rational approximation are analysed. A new method to determine the interpolating rational Fejer – Hermite process is proposed. One of the main results of this paper is the proof of the uniform convergence of this process for an arbitrary function, which is continuous on the segment, under some restrictions for the poles of approximating functions. This result is preceded by some auxiliary statements describing the properties of special rational functions. The classic methods of mathematical analysis, approximation theory, and theory of functions of a complex variable are used to prove the results of the work. Moreover, we present the numerical analysis of the effectiveness of the application of the constructed interpolating Fejer – Hermite process for the approximation of a continuous function with singularities. The choice of parameters, on which the nodes of interpolation depend, is made in several standard ways. The obtained results can be applied to further study the approximating properties of interpolating processes.


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