A general approach to multivariable recursive interpolation

We consider here the problem of constructing a general recursive algorithm to interpolate a given set of data with a rational function. While many algorithms of this kind already exists, they are either providing non minimal-degree solutions (like the Schur algorithm), or exhibit jumps in the degree of the interpolants (or of the partial real- ization, as the problem is called when the interpolation is at infinity, see Rissanen [10] and Gragg-Lindquist [5]). By imbedding the solu- tion into a larger set of interpolants, we show that the increase in the degree of this representation always equals the increase in the length of the data. We provide an algorithm to interpolate multivariable tan- gential sets of data with arbitrary nodes, generalising in a fundamental manner the results of Kuijper in [8].