SINGULAR INTEGRAL OPERATORS IN L2 SPACE WITH CHEBYSHEV WEIGHTS

1IntroductionLetL:=L:l-1,11bethefunctionspacewhichconsist8offunctionsfsatisfyingwherep(t)isaweightfunction,denotethenormthenwecanknowfroml11that.hereexistsanorthogonalsetofpolynodrialsyto,yt1,...,ytnt...andEspeciallyweknowfrom[1]thatifweightfunctionp(f)maybechosenasw1(t)=(1-t')-t,wz(t)=(1-f')i,ws(t)=(1-t)t(1 t)-1,w`(t)=(1-f)-t(1 t)i,thecorrespondingrespectiveorthogonalpolynondalsareTJ')(t)=cosno,Tj')(t)=sin(n 1)o/sino,T1')(t)=sinngo/sint,T;')(f)=cosnge/cos9,wheret=coso,thenby(1.3)theyarethed…     

Asymptotic upper bounds for the coefficients in the Chebyshev series expansion for a general order integral of a function

The usual way to determine the asymptotic behavior of the Chebyshev coefficients for a function is to apply the method of steepest descent to the integral representation of the coefficients. However, the procedure is usually laborious. We prove an asymptotic upper bound on the Chebyshev coefficients for the integral of a function. The tightness of this upper bound is then analyzed for the case , the first integral of a function. It is shown that for geometrically converging Chebyshev series the theorem gives the tightest upper bound possible as . For functions that are singular at the endpoints of the Chebyshev interval, , the theorem is weakened. Two examples are given. In the first example, we apply the method of steepest descent to directly determine (laboriously!) the asymptotic Chebyshev coefficients for a function whose asymptotics have not been given previously in the literature: a Gaussian with a maximum at an endpoint of the expansion interval. We then easily obtain the asymptotic behavior of its first integral, the error function, through the application of the theorem. The second example shows the theorem is weakened for functions that are regular except at . We conjecture that it is only for this class of functions that the theorem gives a poor upper bound.

     

Algorithms for the integration and derivation of Chebyshev series

General formulas for the mth integral and derivative of a Chebyshev polynomial of the first or second kind are presented. The result is expressed as a finite series of the same kind of Chebyshev polynomials. These formulas permit to accelerate the determination of such integrals or derivatives. Besides, it is presented formulas for the mth integral and derivative of finite Chebyshev series and a numerical algorithm for the direct evaluation of the mth derivative of such a series.     

Approximate solution of the problem of scattering of surface water waves by a partially immersed rigid plane vertical barrier

The problem of scattering of two dimensional surface water waves by a partially immersed rigid plane vertical barrier in deep water is re-examined. The associated mixed boundary value problem is shown to give rise to an integral equation of the first kind. Two direct approximate methods of solution are developed and utilized to determine approximate solutions of the integral equation involved. The all important physical quantity, called the Reflection Coefficient, is evaluated numerically, by the use of the approximate solution of the integral equation. The numerical results, obtained in the present work, are found to be in an excellent agreement with the known results, obtained earlier by Ursell (1947), by the use of the closed form analytical solution of the integral equation, giving rise to rather complicated expressions involving Bessel functions.     

On a modification of the Clenshaw-Curtis quadrature formula

In this paper we discuss a modification of the Clenshaw-Curtis quadrature formula. It is shown that for integrals, where the integrand may be expanded in a sufficiently rapid convergent Chebyshev series, we may split the sequence of calculated approximations into two sequences, one which approximates the integral from above and one which approximates it from below. Thus, at any step during the calculation we obtain both upper and lower bounds for the true value of the integral.Work performed while the author was working as a visiting scientist at CERN/Geneve.     

A rapid solution of a kind of 1D Fredholm oscillatory integral equation

How to solve oscillatory integral equations rapidly and accurately is an issue that attracts special attention in many engineering fields and theoretical studies. In this paper, a rapid solution method is put forward to solve a kind of special oscillatory integral equation whose unknown function is much less oscillatory than the kernel function. In the method, an improved-Levin quadrature method is adopted to solve the oscillatory integrals. On the one hand, the employment of this quadrature method makes the proposed method very accurate; on the other hand, only a small number of small-scaled systems of linear equations are required to be solved, so the computational complexity is also very small. Numerical examples confirm the advantages of the method.     

On the Chebyshev type inequality for seminormed fuzzy integral

The Chebyshev type inequality for seminormed fuzzy integral is discussed. The main results of this paper generalize some previous results obtained by the authors. We also investigate the properties of semiconormed fuzzy integral, and a related inequality for this type of integral is obtained.     

A Chebyshev type inequality for Sugeno integral and comonotonicity

We supply a characterization of comonotonicity property by a Chebyshev type inequality for Sugeno integral.     

Numerical solution of the abel integral equation

A numerical method for the solution of the Abel integral equation is presented. The known function is approximated by a sum of Chebyshev polynomials. The solution can then be expressed as a sum of generalized hypergeometric functions, which can easily be evaluated, using a simple recurrence relation.     

Recursive generation of the Galerkin–Chebyshev matrix for convolution kernels

The Galerkin–Chebyshev matrix is the coefficient matrix for the Galerkin method (or the degenerate kernel approximation method) using Chebyshev polynomials. Each entry of the matrix is defined by a double integral. For convolution kernels K(x-y) on finite intervals, this paper obtains a general recursion relation connecting the matrix entries. This relation provides a fast generation of the Galerkin–Chebyshev matrix by reducing the construction of a matrix of order N from N ²+O(N) double integral evaluations to 3N+O(1) evaluations. For the special cases (a) K(x-y)=|x-y|^α-1(-ln|x-y|) ^p and (b) K(x-y)=K _ν(σ|x-y|) (modified Bessel functions), the number of double integral evaluations to generate a Galerkin–Chebyshev matrix of arbitrary order can be further reduced to 2p+2 double integral evaluations in case (a) and to 8 double integral evaluations in case (b). This revised version was published online in June 2006 with corrections to the Cover Date.     

Chebyshev type inequality for Choquet integral and comonotonicity

We supply a Chebyshev type inequality for Choquet integral and link this inequality with comonotonicity.     

Multivariate inequalities based on Sobolev representations

Here we derive very general multivariate tight integral inequalities of Chebyshev–Grüss, Ostrowski types and of comparison of integral means. These are based on well-known Sobolev integral representation of a function. Our inequalities engage ordinary and weak partial derivatives of the involved functions. We also give their applications. On the way to prove our main results we derive important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. Our results expand to all possible directions.     

γ-条件下变形Chebyshev迭代方法的收敛性及其应用

在sm a le点估计理论引导下,利用优序列方法,研究γ-条件下,变形chebyshev迭代方法在求解Banach空间中非线性方程F(x)=0时的收敛性问题,并给出了误差估计,而且通过一个积分方程实例比较了它和N ew ton法,导数超前计值的变形N ew ton法,避免导数求逆的变形N ew ton法的每步误差.     

A Method for Solving Fredholm Integral Equations of the First Kind Based on Chebyshev Wavelets

In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.     

On an improved-Levin oscillatory quadrature method

Performance of an improved-Levin quadrature method for oscillatory integrals is studied. In the study, the behavior of the target system of linear equations is analyzed and an error reduction factor is proposed to measure the behavior?s impact on the integral result. Numerical investigations show that the error reduction factor is extremely small for ill-conditioned case, and the ill-conditioning has little impact on the final integral result. Therefore, the concerned quadrature method is numerically very stable and it has addressed the Levin method?s problem of being susceptible to the ill-conditioning.     

Adaptive Bivariate Chebyshev Approximation

We propose an adaptive algorithm which extends Chebyshev series approximation to bivariate functions, on domains which are smooth transformations of a square. The method is tested on functions with different degrees of regularity and on domains with various geometries. We show also an application to the fast evaluation of linear and nonlinear bivariate integral transforms.     

Operational matrices of Chebyshev cardinal functions and their application for solving delay differential equations arising in electrodynamics with error estimation

In this paper, a new and effective direct method to determine the numerical solution of pantograph equation, pantograph equation with neutral term and Multiple-delay Volterra integral equation with large domain is proposed. The pantograph equation is a delay differential equation which arises in quite different fields of pure and applied mathematics, such as number theory, dynamical systems, probability, mechanics and electrodynamics. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration, product and delay of the Chebyshev cardinal functions are presented. A general procedure for forming these matrices is given. These matrices play an important role in modelling of problems. By using these operational matrices together, a pantograph equation can be transformed to a system of algebraic equations. An efficient error estimation for the Chebyshev cardinal method is also introduced. Some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results.     

Complete Hyper-elliptic Integrals of the First Kind and the Chebyshev Property

This paper is devoted to study the following complete hyper-elliptic integral of the first kind $$J(h)=\oint\limits_{\Gamma_h}\frac{\alpha_0+\alpha_1x+\alpha_2x^2+\alpha_3x^3}{y}dx,$$ where $\alpha_i\in\mathbb{R}$, $\Gamma_h$ is an oval contained in the level set $\{H(x,y)=h, h\in(-\frac{5}{36},0)\}$ and $H(x,y)=\frac{1}{2}y^2-\frac{1}{4}x^4+\frac{1}{9}x^9$. We show that the 3-dimensional real vector spaces of these integrals are Chebyshev for $\alpha_0=0$ and Chebyshev with accuracy one for $\alpha_i=0\ (i=1,2,3)$.