Hermite Paratrigonometric Interpolation Formulae And Their Applications

There are substantial relation between the interpolation and approximation of functions. The interpolation polynomial is regarded as the important tool of achieving approximation of function. It is always applied for deducing numerical differential and integral, and looking for numerical solutions of differential or integral equations. The study on the algebraic interpolation and the trigonometric interpolation are rapidly developed in view of both theoretical reasons and applications. The trigonometric interpolation and its basis play the essential role on calculating numerical integral of periodic function and numerical solutions of singular integral equations with periodic functions. The difficulty on the theory of trigonometric interpolation is more than the algebraic ones and even the manners to investigate them are different from each other.The Hermite trigonometric interpolation problem with different multiplicities became an interesting and flourish subject in the last twenty years. Many mathematicians are working on it and they obtained enormous achievement by using different methods. In 1960, H. E. Salzer first discussed the Hermite trigonometric Interpolation for non-equidistant interpolation points with uniform multiplicities. His result was only concerned with the interpolation basis without any analysis for the remainder term. In 1972, R. Kress studied the Hermite trigonometric interpolation problem for even number of equidistant interpolation points with uniform multiplicities, which has generalized the Salzer's result. In this article, he defined a class of special trigonometric "original" functions with nice properties. Based on these functions, Kress established the basis of trigonometric interpolation and obtained the Hermite trigonometric interpolation formulas. Meantime he discussed the convergence of Hermite trigonometric interpolation and its application to numerical integral. Most succedent discussions on Hermite trigonometric interpolation are mainly based on the ideas and tools in which he introduced for the explicit representations for the fundamental Hermite polynomials and the remainder term. Tom Lyche and I. I chim studied this problem by using the method of trigonometric divided differences in 1979 and 1983, respectively, and the former gave the Newton form of Hermite interpolation polynomials. In 1993, F. J. Delvos introduced the π-periodic function and π-antiperiodic function. HeIIIconsidered the -periodic and -antiperiodic Hermite trigonometric interpolation, respectively, for arbitrary odd and even number of interpolation points with different multiplicities lying on (0,7r). The discussion of these functions is much more easier than the 27r-periodic Hermite trigonometric interpolation. It is regretful that he has also not given any analysis of the remainder term, and on the other hand, those trigonometric interpolation polynomials he obtained are not in the minimum order. In other words, they are the Hermite trigonometric interpolation polynomials with the understanding under the scheme of the 7r-translation nodes. In 1994, D. P. Dryanov proved the existence and uniqueness of the Hermite trigonometric interpolation polynomial by the method based on the Chebyshev system for general cases, i.e. arbitrary number of interpolation points and multiplicities However, to get the constructive fundamental Hermite trigonometric polynomials is very still difficult by applying for this method. In 1997, Jin Guo-xiang constructed the fundamental Hermite polynomials for the general case by combining the methods of Kress and Delvos with the definition of the trigonometric polynomial H(a).The quadrature formulae of proper integrals for any number of interpolation points with different multiplicities have also been investigated by many mathematicians, as the natural application of the Hermite trigonometric interpolation with different multiplicities. Such research is more difficult than the quadrature formulae of periodic function integrals for any number of interpolation points with 1-ord...