The Research Of Quadrature Formule For Singular Integrals With Cauchy Kernel Based On Trigonometric Interpolation

Fundamental study on several quadrature formulas and quadrature error estimates for singular integrals with Cauchy kernel based on trigonometric interpolation, and the convergence of the quadrature formula for singular integrals with Cauchy kernel with Legendre weight are discussed in this thesis which consists of three parts.In the first part, the definition of singular integrals with Cauchy kernel and some relational lemmas with their proofs are introduced.In the second part, the quadrature problem for singular integrals with Cauchy kernel with Legendre weight is discussed by the method of trigonometric interpolation. Firstly, the trigonometric form of singular integrals with Cauchy kernel is obtained by trigonometric transform, and the quadrature formula of singular integrals with Cauchy kernel is established by cosine polynomials approximate density function; secondly, the error estimate and error bound are analysed; thirdly, the convergence of the quadrature formula for singular integrals with Cauchy kernel with Legendre weight is dicussed.In the third part, the quadrature problems for singular integrals with Cauchy kernel with some special weights are dicussed by the method of trigonometric interpolation. According to the expression of different weight function, the quadrature formula is not the same. Firstly, the quadrature problem for singular integrals with Cauchy kernel with the ultraspherical weight function is discussed. The automatic quadrature formula is proposed by trigonometric transform and cosine polynomials, which is independent of the set of values of poles, and the error estimate between the quadrature formula and the singular integrals with Cauchy kernel with the ultraspherical weight function is discussed; secondly, the quadrature problem for singular integrals with Cauchy kernel with Jacobi weight function is discussed. The automatic quadrature formula is proposed, which is independent of the set of values of poles as the case of Legendre and ultraspherical weight function, and the error estimate is given.