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 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 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 Analysis of Numerical Measure and Numerical Integration Based on Measure

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Yinkun Wang ◽  
Jianshu Luo ◽  
Xiangling Chen
Keyword(s):  
Numerical Method ◽  
Convergence Analysis ◽  
Numerical Integration ◽  
Error Bound ◽  
Lagrange Interpolation ◽  
Singular Integrals ◽  
Specific Method ◽  
Numerical Examples ◽  
Measure Function ◽  
Theoretical Results

We present a convergence analysis for a general numerical method to estimate measure function. By combining Lagrange interpolation, we propose a specific method for approximating the measure function and analyze the convergence order. Further, we analyze the error bound of numerical measure integration and prove that the numerical measure integration can decrease the singularity for singular integrals. Numerical examples are presented to confirm the theoretical results.

scholarly journals Asymptotic Analysis and Error Estimate for Rosenau-Burgers Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Jun Zhang ◽  
Zixin Liu ◽  
Fubiao Lin ◽  
Jianjun Jiao
Keyword(s):  
Error Estimate ◽  
Numerical Method ◽  
Numerical Scheme ◽  
Burgers Equation ◽  
Stability Result ◽  
Unconditionally Stable ◽  
Numerical Examples ◽  
Polynomial Degree ◽  
Forcing Function ◽  
Theoretical Results

In this work, the asymptotic stability result for Rosenau-Burgers equation is established, under appropriate assumptions on steady state eigenvalue problem and the forcing function. In addition, we propose and analyze a linearized numerical method for solving this nonlinear Rosenau-Burgers equation. We prove that the numerical scheme is unconditionally stable, and the error estimate shows that the numerical method is in the order of O(Δt2+N2-m), where Δt, N, and m are, respectively, step of time, polynomial degree, and regularity of u. Numerical examples are illustrated to verify the theoretical results.

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.

scholarly journals Numerical method of identification of an unknown source term in a heat equation

2002 ◽  
Vol 8 (2) ◽  
pp. 161-168 ◽  
Author(s):  
Afet Golayoğlu Fatullayev
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Numerical Method ◽  
Minimization Problem ◽  
Unknown Function ◽  
Source Term ◽  
Numerical Procedure ◽  
Numerical Examples ◽  
Unknown Source

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. Numerical examples are presented.

Convergence and approximate calculation of average degree under different network sizes for decreasing random birth-and-death networks

2018 ◽  
Vol 32 (15) ◽  
pp. 1850159
Author(s):  
Yin Long ◽  
Xiao-Jun Zhang ◽  
Kui Wang
Keyword(s):  
Analytical Solution ◽  
Numerical Method ◽  
Power Law ◽  
Approximate Calculation ◽  
Approximate Expression ◽  
Network Size ◽  
Average Degree ◽  
Large Network ◽  
Network Link ◽  
Theoretical Results

In this paper, convergence and approximate calculation of average degree under different network sizes for decreasing random birth-and-death networks (RBDNs) are studied. First, we find and demonstrate that the average degree is convergent in the form of power law. Meanwhile, we discover that the ratios of the back items to front items of convergent reminder are independent of network link number for large network size, and we theoretically prove that the limit of the ratio is a constant. Moreover, since it is difficult to calculate the analytical solution of the average degree for large network sizes, we adopt numerical method to obtain approximate expression of the average degree to approximate its analytical solution. Finally, simulations are presented to verify our theoretical results.

An L∞-error Estimate for the h-p Version Continuous Petrov-Galerkin Method for Nonlinear Initial Value Problems

2015 ◽  
Vol 5 (4) ◽  
pp. 301-311 ◽  
Author(s):  
Lijun Yi
Keyword(s):  
Error Estimate ◽  
Galerkin Method ◽  
Error Bound ◽  
Initial Value Problems ◽  
Initial Value ◽  
Numerical Examples ◽  
Time Stepping ◽  
Theoretical Results

AbstractThe h-p version of the continuous Petrov-Galerkin time stepping method is analyzed for nonlinear initial value problems. An L∞-error bound explicit with respect to the local discretization and regularity parameters is derived. Numerical examples are provided to illustrate the theoretical results.

scholarly journals More Low Differential Uniformity Permutations over F22k with k Odd

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yue Leng ◽  
Jinyang Chen ◽  
Tao Xie
Keyword(s):  
Block Ciphers ◽  
Algebraic Degree ◽  
Differential Uniformity ◽  
Numerical Examples ◽  
High Nonlinearity ◽  
Substitution Boxes ◽  
Theoretical Results

Permutations with low differential uniformity, high algebraic degree, and high nonlinearity over F22k can be used as the substitution boxes for many block ciphers. In this paper, several classes of low differential uniformity permutations are constructed based on the method of choosing two permutations over F22k to get the desired permutations. The resulted low differential uniformity permutations have high algebraic degrees and nonlinearities simultaneously, which provide more choices for the substitution boxes. Moreover, some numerical examples are provided to show the efficacy of the theoretical results.

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