In this paper, we present a new method of interpolation on equally spaced collocation points in a jixed interval on the x-axis. 7'he proposed scheme is an iterative one which combines Taylor expansion and Cauchy's integral representation. The conuergence rate of the method is investigated theoretically and verified numerically. Additionally, some numerical results are also shown to discuss the usefwlness of the. present method.
I. Infroduciion
Up to the present, various types of interpolation formulae have been known, such as Chebyshev expansions, trigonometric interpolations (or finite Fourier transforms) and spline expansions. They have their own characteristics, advantages and disadvantages, as is generally described in (1) and (2).
In this paper, we shall derive a new iterative interpolation formula which has utterly different features from the others on the basis of Cauchy's integral representation which employs the first-order Taylor series around each collocation point. Since this technique assumes the analytic property of the function from which data are sampled, the data should be sufhciently smooth. This seems to restrict somewhat the availability of the method. However, it can be considered that computer simulations in engineering problems frequently produce smooth data. Therefore, the method does not lose its practicality.
More specit%Aly, the background of the present method can be understood as follows: From practical points of view, so long as we restrict ourselves to the case of functions usually occurring in physical problems, most of them are analytic. Therefore, it is acceptable to incorporate the analytic property in the interpolation formula. Thus, in the present method, Cauchy's integral representation plays an important role, similarly as in (3), (4).
II. heliminaly
Consider a real analytic function f(x) defined in the closed interval K =
[ -1, l] on the real x-axis. And let the corresponding complex function f(z) be @The Fkankh Institute 0016_0032/81/050323_08S02,w/w