On the osculatory rational interpolation problem

Abstract

The problem of the existence and construction of a table of osculating rational functions r 1 , m {r_{1,m}} for 1 , m ⩾ 0 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 {s_i} - 1 at points x i {x_i} for i ⩾ 0 i \geqslant 0 . Using these results, we prove the existence of continued fractions of the form \[ c 0 + c 1 ⋅ ( x − y 0 ) + … + c k ⋅ ( x − y 0 ) … ( x − y k − 1 ) + c k + 1 ⋅ ( x − y 0 ) … ( x − y k ) 1 + c k + 2 ⋅ ( x − y k + 1 ) 1 + c k + 3 ⋅ ( x − y k + 2 ) 1 + … , {c_0} + {c_1} \cdot (x - {y_0}) + \ldots + {c_k} \cdot (x - {y_0}) \ldots (x - {y_{k - 1}}) + \frac {{{c_{k + 1}} \cdot (x - {y_0}) \ldots (x - {y_k})}}{1} + \frac {{{c_{k + 2}} \cdot (x - {y_{k + 1}})}}{1} + \frac {{{c_{k + 3}} \cdot (x - {y_{k + 2}})}}{1} + \ldots , \] with the y k {y_k} suitably selected from among the x i {x_i} , whose convergents form the elements r k , 0 , r k + 1 , 0 , r k + 1 , 1 , r k + 2 , 1 , … {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 {c_i} for i ⩾ 0 i \geqslant 0 . This algorithm is a generalization of the qd-algorithm.