The Researches On Quadrature Formulas With Multiple Nodes And Singular Integrals With Hilbert Kernel

In this thesis, we discuss the fundamental study on the problem ofquadrature formula of Gaussian type for proper integrals of periodic functionswith multiple nodes and quadrature formula of Gaussian type for singularintegrals with Hilbert kernel and simple nodes. The thesis consists of threeparts.In the first part, we consider the quasi-orthogonal trigonometricpolynomials with respect to a weight function. Firstly, the concepts andlemmas of the (para)trigonometric polynomial and its family are introduced.Secondly, we give the definition of quasi-orthogonal trigonometricpolynomials. It is proved that the quasi-orthogonal trigonometric polynomialsof degreen2has exactly n distinct simple zeros in [,). In addition, theproblem of existence and uniqueness of quasi-orthogonal trigonometricpolynomials is considered.In the second part, we establish quadrature formula of Gaussian type forproper integrals of periodic functions with multiple nodes by using thequasi-orthogonal trigonometric polynomials. First of all, we present aniterative method for calculating quadrature nodes, which is also the zero ofquasi-orthogonal trigonometric polynomials. Secondly, using thetrigonometric degree of exactness, we construct some systems of linearequations, so that the problem of calculation of quadrature weights is reducedto solving the systems of linear equations with unknown quadrature weights,which can be decomposed to upper triangular systems. Solving this systems,we obtain the quadrature weights with even nodes. It has completedMlinovanovic’s result. Lastly, we derive quadrature formula of H TM()type with multiple nodes, which is also a new result.In the third part, we discuss the problem of quadrature formula ofGaussian type for singular integrals with Hilbert kernel and simple nodes.First of all, we construct quadrature nodes of quadrature formula of Gaussiantype for proper integrals of periodic functions with simple nodes, and weprove that the orthogonal (para)trigonometric polynomials with respect to aweight function satisfy a three-term recurrence relation in every family oftrigonometric polynomials. Moreover, we present an iterative method forcalculating quadrature nodes by the method of translating real orthogonal(para)trigonometric polynomials into complex algebric polynomials.Secondly, we establish the quadrature formula of Gaussian type for singularintegrals with Hilbert kernel and simple nodes by the method of separation ofsingularities. In this part, we solve the problem of constructing the quadraturenodes of quadrature formula for singular integrals with Hilbert kernel.