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 exist, 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 realization, as the problem is called when the interpolation is at infinity, see Rissanen (SIAM J Control 9(3):420-430, 1971) and Gragg and Lindquist (in: Linear systems and control (special issue), linear algebra and its applications, vol 50. pp 277-319, 1983)). By imbedding the solution into a larger set of interpolants, we show that the increase in the degree of this representation is proportional to the increase in the length of the data. We provide an algorithm to interpolate multivariable tangential sets of data with arbitrary nodes, generalizing in a fundamental manner the results of Kuijper (Syst Control Lett 31:225-233, 1997). We use this new approach to discuss a special scalar case in detail. When the interpolation data are obtained from the Taylor-series expansion of a given function, then the Euclidean-type algorithm plays an important role.