scholarly journals Trigonometric multiple orthogonal polynomials of semi-integer degree and the corresponding quadrature formulas

2014 ◽  
Vol 96 (110) ◽  
pp. 211-226 ◽  
Author(s):  
Gradimir Milovanovic ◽  
Marija Stanic ◽  
Tatjana Tomovic
Keyword(s):  
Orthogonal Polynomials ◽  
Trigonometric Polynomials ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
Numerical Examples ◽  
Multiple Orthogonal Polynomials ◽  
Optimal Set ◽  
Numer Math ◽  
Theoretical Results

An optimal set of quadrature formulas with an odd number of nodes for trigonometric polynomials in Borges? sense [Numer. Math. 67 (1994), 271-288], as well as trigonometric multiple orthogonal polynomials of semi-integer degree are defined and studied. The main properties of such a kind of orthogonality are proved. Also, an optimal set of quadrature rules is characterized by trigonometric multiple orthogonal polynomials of semiinteger degree. Finally, theoretical results are illustrated by some numerical examples.

scholarly journals Multiple orthogonality in the space of trigonometric polynomials of semi-integer degree

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2227-2237
Author(s):  
Marija Stanic ◽  
Tatjana Tomovic
Keyword(s):  
Numerical Method ◽  
Recurrence Relations ◽  
Trigonometric Polynomials ◽  
Quadrature Rules ◽  
Numerical Examples ◽  
Optimal Set ◽  
Numer Math ◽  
Theoretical Results

In this paper we consider multiple orthogonal trigonometric polynomials of semi-integer degree, which are necessary for constructing of an optimal set of quadrature rules with an odd number of nodes for trigonometric polynomials in Borges? sense [Numer. Math. 67 (1994) 271-288]. We prove that such multiple orthogonal trigonometric polynomials satisfy certain recurrence relations and present numerical method for their construction, as well as for construction of mentioned optimal set of quadrature rules. Theoretical results are illustrated by some numerical examples.

scholarly journals Generalized Gaussian quadratures for integrals with logarithmic singularity

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 1111-1126 ◽  
Author(s):  
Gradimir Milovanovic
Keyword(s):  
Logarithmic Singularity ◽  
Gaussian Quadrature ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
Short Account ◽  
Numerical Examples ◽  
Gaussian Quadrature Formulas ◽  
Gaussian Quadratures ◽  
Logarithmic Singularities

A short account on Gaussian quadrature rules for integrals with logarithmic singularity, as well as some new results for weighted Gaussian quadrature formulas with respect to generalized Gegenbauer weight x |? |x|(1-x2)?, ? > -1, on (-1,1), which are appropriated for functions with and without logarithmic singularities, are considered. Methods for constructing such kind of quadrature formulas and some numerical examples are included.

scholarly journals Orthogonal polynomials and generalized Gauss-Rys quadrature formulae

2021 ◽  
Vol 49 (1) ◽  
Author(s):  
Gradimir V. Milovanovic ◽  
◽  
Nevena Vasovic ◽  
Keyword(s):  
Orthogonal Polynomials ◽  
Quadrature Rules ◽  
Quadrature Formulas ◽  
Numerical Construction ◽  
Quadrature Formulae ◽  
Gaussian Type ◽  
Modified Moments

Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type concerni λ ng > the 1 e / v 2 en wei x gh > t f 0 unction ω(t; x) = exp λ (−= xt 1 2) / ( 2 1 − t2)−1/2 on (−1, 1), with parameters − and , are considered. For these quadrature rules reduce to the socalled Gauss-Rys quadrature formulas, which were investigated earlier by several authors, e.g., Dupuis at al 1976 and 1983; Sagar 1992; Schwenke 2014; Shizgal 2015; King 2016; Milovanovic ´ 2018, etc. In this generalized case, the method of modified moments is used, as well as a transformation of quadratures on (−1, 1) with N nodes to ones on (0, 1) with only (N + 1)/2 nodes. Such an approach provides a stable and very efficient numerical construction.

scholarly journals Quadrature rules with an even number of multiple nodes and a maximal trigonometric degree of exactness

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2239-2255 ◽  
Author(s):  
Tatjana Tomovic ◽  
Marija Stanic
Keyword(s):  
Numerical Method ◽  
Trigonometric Polynomials ◽  
Quadrature Rules ◽  
Numerical Examples ◽  
Degree Of Exactness ◽  
Interpolatory Quadrature

This paper is devoted to the interpolatory quadrature rules with an even number of multiple nodes, which have the maximal trigonometric degree of exactness. For constructing of such quadrature rules we introduce and consider the so-called s- and ?-orthogonal trigonometric polynomials. We present a numerical method for construction of mentioned quadrature rules. Some numerical examples are also included.

scholarly journals Some high-order zero-finding methods using almost orthogonal polynomials

1975 ◽  
Vol 19 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Richard P. Brent
Keyword(s):  
Orthogonal Polynomials ◽  
Order Of Convergence ◽  
Function Evaluation ◽  
Order Method ◽  
Order Zero ◽  
Numerical Examples ◽  
Zeros Of Functions ◽  
Multipoint Iterative Methods ◽  
Fourth Order Method ◽  
Theoretical Results

AbstractSome multipoint iterative methods without memory, for approximating simple zeros of functions of one variable, are described. For m > 0, n ≧ 0, and k satisfying m + 1 ≧ k > 0, there exist methods which, for each iteration, use one evaluation of f, f′, … f(m) followed by n evaluations of f(k), and have order of convergence m + 2n + 1. In particular, there are methods of order 2(n + 1) which use one function evaluation and n + 1 derivative evaluations per iteration. These methods naturally generalize the known cases n = 0 (Newton's method) and n = 1 (Jarratt's fourth-order method), and are useful if derivative evaluations are less expensive than function evaluations. To establish the order of convergence of the methods we prove some results, which may be of independent interest, on orthogonal and “almost orthogonal” polynomials. Explicit, nonlinear, Runge-Kutta methods for the solution of a special class of ordinary differential equations may be derived from the methods for finding zeros of functions. The theoretical results are illustrated by several numerical examples.

A total positivity property of the Marchenko-Pastur Law

2015 ◽  
Vol 30 ◽  
Author(s):  
Ana Marco ◽  
Jose-Javier Martinez
Keyword(s):  
Orthogonal Polynomials ◽  
Gaussian Quadrature ◽  
Total Positivity ◽  
Quadrature Formulas ◽  
Positivity Property ◽  
Accurate Computation ◽  
Gaussian Quadrature Formulas ◽  
Theoretical Results

A property of the Marchenko-Pastur measure related to total positivity is presented. The theoretical results are applied to the accurate computation of the roots of the corresponding orthogonal polynomials, an important issue in the construction of Gaussian quadrature formulas.

scholarly journals Strong asymptotics for the Pollaczek multiple orthogonal polynomials

2015 ◽  
Vol 92 (3) ◽  
pp. 709-713
Author(s):  
A. I. Aptekarev ◽  
G. López Lagomasino ◽  
A. Martínez-Finkelshtein
Keyword(s):  
Orthogonal Polynomials ◽  
Strong Asymptotics ◽  
Multiple Orthogonal Polynomials

Multiple Orthogonal Polynomials on the Unit Circle

2007 ◽  
Vol 28 (2) ◽  
pp. 173-197 ◽  
Author(s):  
Judit Minguez Ceniceros ◽  
Walter Van Assche
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Multiple Orthogonal Polynomials

scholarly journals Edge detection with trigonometric polynomial shearlets

2021 ◽  
Vol 47 (1) ◽  
Author(s):  
Kevin Schober ◽  
Jürgen Prestin ◽  
Serhii A. Stasyuk
Keyword(s):  
Edge Detection ◽  
Trigonometric Polynomial ◽  
Two Dimensions ◽  
Characteristic Functions ◽  
Numerical Examples ◽  
Special Cases ◽  
Periodic Characteristic ◽  
Boundary Curves ◽  
Theoretical Results ◽  
De La Vallée Poussin

AbstractIn this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.

scholarly journals Stability Analysis of Impulsive Control Systems with Finite and Infinite Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Xuling Wang ◽  
Xiaodi Li ◽  
Gani Tr. Stamov
Keyword(s):  
Control Systems ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Delay Systems ◽  
Stability Criteria ◽  
Numerical Examples ◽  
Infinite Delays ◽  
Impulsive Control Systems ◽  
Stable Matrix ◽  
Theoretical Results

This paper studies impulsive control systems with finite and infinite delays. Several stability criteria are established by employing the largest and smallest eigenvalue of matrix. Our sufficient conditions are less restrictive than the ones in the earlier literature. Moreover, it is shown that by using impulsive control, the delay systems can be stabilized even if it contains no stable matrix. Finally, some numerical examples are discussed to illustrate the theoretical results.

Sign in / Sign up

Export Citation Format

Share Document