On the osculatory rational interpolation problem

Abstract:

The problem of the existence and construction of a table of osculating rational functions ${r_{1,m}}$ for $1,m \geqslant 0$ is considered. First, a survey is given of some results from the theory of osculatory rational interpolation of order ${s_i} - 1$ at points ${x_i}$ for $i \geqslant 0$. Using these results, we prove the existence of continued fractions of the form \[ {c_0} + {c_1} \cdot (x - {y_0}) + \ldots + {c_k} \cdot (x - {y_0}) \ldots (x - {y_{k - 1}}) + \frac {{{c_{k + 1}} \cdot (x - {y_0}) \ldots (x - {y_k})}}{1} + \frac {{{c_{k + 2}} \cdot (x - {y_{k + 1}})}}{1} + \frac {{{c_{k + 3}} \cdot (x - {y_{k + 2}})}}{1} + \ldots ,\] with the ${y_k}$ suitably selected from among the ${x_i}$, whose convergents form the elements ${r_{k,0}},{r_{k + 1,0}},{r_{k + 1,1}},{r_{k + 2,1}}, \ldots$ of the table. The properties of these continued fractions make it possible to derive an algorithm for constructing their coefficients ${c_i}$ for $i \geqslant 0$. This algorithm is a generalization of the qd-algorithm.