On the history of multivariate polynomial interpolation

Multivariate polynomial interpolation is a basic and fundamental subject in Approximation Theory and Numerical Analysis, which has received and continues receiving not deep but constant attention. In this short survey we review its development in the rst 75 years of this century, including a pioneering paper by Kronecker in the 19th century. x1. Introduction Interpolation, by polynomials or other functions, is a rather old method in applied mathematics. This is already indicated by the fact that, apparently, the word \interpolation" itself has been introduced by J. Wallis as early as 1655 as it is claimed in 13]. Compared to this, polynomial interpolation in several variables is a relatively new topic and probably only started in the second half of the last century with work by W. Borchardt 6] and L. Kro-necker 22]. If one considers, for example, the Encyklopp adie der Mathema-tischen Wissenschaften 13] (Encyclopedia of Math. Sciences), originated by the Preuuische Akademie der Wissenschaften (Prussian Academy of Sciences) to sum up the \state of art" of mathematics at its time, then the part on interpolation, written by J. Bauschinger (Bd. I, Teil 2), mentions only one type of multivariate interpolation, namely (tensor) products of sine and cosine functions in two variables, however, without being very speciic. The French counterpart, the Encyclop edie de Sciences Mathematiques 14], also contains a section on interpolation (Tome I, vol. 4), where H. Andoyer translated and extended Bauschinger's exposition. Andoyer is even more explicit with his opinion on multivariate polynomial interpolation, by making the following statement which we think that time has contradicted: Il est manifeste que l'interpolation des fonctions de plusiers variables ne demande aucun principe nouveau, car dans tout ce qui pr ec ede le fait que la variable ind ependante etait unique n'a souvent jou e aucun r^ ole. 1 This is just a preprint by 1 M. Gasca and T. Sauer pp. 1{3. 1 It is clear that the interpolation of functions of several variables does not demand any new principles because in the above exposition the fact that the variable was unique has not played frequently any role.