Contributions to the problem of approximation of equidistant data by analytic functions. Part B. On the problem of osculatory interpolation. A second class of analytic approximation formulae

1 The connection of these types characteristics with the notation as used by Greville in his paper The general theory of osculatory interpolation, Trans. Actuar. Soc. Amer., 45, pp. 202-265 (1944), especially pp. 210-211, is as follows: The first three symbols D~, C, El, require no further comment since they are identical respectively with the characteristics 4, 1, and 6 of Greville's classification, pp. 210-211. There remain three further characteristics to be discussed: (i) Whether the formula (1) is an "end-point" or "mid-point" formula. This point is of importance if (1) is written in terms of certra! differences, since it, then reduces to either the Everett or else the Steffensen form. The following statement is obvious: The formula (1) is an "end point" or "mid-point" formula depending on whether the span s is even or odd.