The Research Of Lacunary Quadrature For Singular Integrals With Hilbert Kernel

Fundamental study for the type of ( 0,p (D)) Birkhoff (para)trigonometric interpolation and Lacunary quadrature for singular integrals with Hilbert kernel is discussed in this thesis which consists of three parts.In the first part,the type of ( 0,p (D)) Birkhoff (para)trigomometric interpolation problem with equidistant nodes on [ 0,2π) is discussed. First of all,the concepts and lemmas of trigonometric polynomial and its kind are introduced. Then,the solvable conditions which are necessary and surfficient to the type of ( 0,p (D)) Birkhoff trigonometric interpolation problem in the kind of the general trigonometric interpolation are given and the explicit expressions of the interpolation basis are constructed. The bases established respectively by Sharma,Varma,Sun Xiehua and so on are special cases of this thesis. Last,the solvable conditions which are necessary and surfficient to the type of ( 0,p (D)) Birkhoff paratrigonometric interpolation problem in the kind of the general trigonometric interpolation are given and the explicit expressions of the interpolation basis are constructed.In the second part,we discussed the problem of lacunary interpolation quadrature for proper integrals of periodic functions. The lacunary quadrature formula of proper integrals is established by the type of ( 0,p (D)) Birkhoff trigonometric interpolation. We discuss the trigonometric precision of the interpolation quadrature formula,and so does the estimate of the remainder. The lacunary quadrature formula established by Varma is special cases here.In the third part,the problem of lacunary quadrature for singular integrals with Hilbert kernel is researched. The lacunary quadrature formula for singular integrals with Hilbert kernel is established by the method of separation of singularities and the lacunary quadrature formula of proper integrals,and the trigonometric precision of the quadrature formula is established.