User:DVD206/Zolotarev problem

The following rational approximation problem demonstrates how fast the considered discrete models converge to their continuous analogs. The problem of optimal rational approximation of the square root on an interval was solved by Russian mathematician Zolotarev with the closed form solutions in terms of elliptic integrals, see [11] & [26].

The elliptic function

$\xi =sn(u,k)$

can be defined by the following integral equation:

$u=\int _{0}^{\xi }{\frac {dt}{\sqrt {(1-t^{2})(1-\kappa ^{2}t^{2})}}}.$

Let

$0<\kappa <1{\mbox{ and }}\kappa '={\sqrt {1-\kappa ^{2}}}$

The optimal rational approximation function of order (k-1,k).

$||1-{\sqrt {\lambda }}{\tilde {r}}(\lambda )||_{C[1,1/\kappa '^{2}]}=inf\{||1-{\sqrt {\lambda }}{r}(\lambda )||_{C[1,1/\kappa '^{2}]},r\in R_{k-1,k}\}$

is given by

${\tilde {r}}(\lambda )=c{\frac {\prod _{l=1}^{k-1}(\lambda +c_{2l})}{\prod _{l=1}^{k}(\lambda +c_{2l-1})}},$

where

$c_{l}={\frac {sn^{2}(lk/(2K),\kappa )}{1-sn^{2}(lk/(2K),\kappa )}},l=1,2,\ldots ,2k-1.$

and

$K=\int _{0}^{1}{\frac {dt}{\sqrt {(1-t^{2})(1-\kappa ^{2}t^{2})}}}$

Exercise (***). Use the interlacing of zeros property to prove that the function

$s(\lambda )=\lambda {\tilde {r}}(\lambda ^{2})$

can be written as a Stieltjes continued fraction.

It was proved in [11] that the error of the optimal approximation converges exponentially fast with the increase in the degree of the rational function.