Interpolation of functions by continued fraction and its generalizations in the case of functions of several variables (Q2730560)

The author proposes a new method of interpolation of a function \(f\) of three real variables by a three-dimensional interpolation continued fraction of the form NEWLINE\[NEWLINED_{n}(x,y,z)=F_0^{(n)}+\mathop{\text \mathbf K}\limits_{k=1}^{n} {(x-x_{k-1})(y-y_{k-1})(z-z_{k-1})\over F_{k}^{(n)}(x,y,z)}, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\displaystyle\mathop{\text \mathbf K}\limits_{k=1}^{n} {\displaystyle a_{k}\over\displaystyle b_{k}}= {\displaystyle a_{1}\over {\displaystyle b_{1}+{\strut a_2\over b_2\ldots +{{a_{n}}\over{b_{n}}}}}},NEWLINE\]NEWLINE \(F_{k}^{(n)}(x,y,z)\) are some combinations of one-dimensional and two-dimensional continued fractions. All continued fractions are determined by the condition NEWLINE\[NEWLINED_{n}(x_{i},y_{j},z_{l})=f(x_{i},y_{j},z_{l})= c_{i,j,l},\;i,j,l=0,\ldots,nNEWLINE\]NEWLINE and by the given points \((x_{i},y_{j},z_{l})\).