Zolotarev polynomials

The original solution to the approximation problem given by Zolotarev was in terms of Jacobi elliptic functions. Zolotarev gave the general solution where the number of zeroes to the left of the peak value (${\displaystyle q}$) in the interval ${\displaystyle [-1,1]}$ is not equal to the number of zeroes to the right of this peak (${\displaystyle p}$). The degree of the polynomial is ${\displaystyle n=p+q}$. For many applications, ${\displaystyle p=q}$ is used and then only ${\displaystyle n}$ need be considered. The general Zolotarev polynomials are defined as[5]

${\displaystyle Z_{n}(x|\kappa )={\frac {(-1)^{p}}{2}}\left[\left({\dfrac {H(u-v)}{H(u+v)}}\right)^{n}+\left({\dfrac {H(u+v)}{H(u-v)}}\right)^{n}\right]}$

where

${\displaystyle u=F\left(\left.\operatorname {sn} \left(\left.v\right|\kappa \right){\sqrt {\dfrac {1+x}{x+2\operatorname {sn} ^{2}\left(\left.v\right|\kappa \right)-1}}}\right|\kappa \right)}$

${\displaystyle v={\dfrac {p}{n}}K(\kappa )}$

${\displaystyle H(\varphi )}$ is the Jacobi eta function

${\displaystyle F(\varphi |\kappa )}$ is the incomplete elliptic integral of the first kind

${\displaystyle K(\kappa )}$ is the quarter-wave complete elliptic integral of the first kind. That is, ${\displaystyle K(\kappa )=F\left(\left.{\frac {\pi }{2}}\right|\kappa \right)}$[6]

${\displaystyle \kappa }$ is the Jacobi elliptic modulus

${\displaystyle \operatorname {sn} (\varphi |\kappa )}$ is the Jacobi elliptic sine.

The variation of the function within the interval [−1,1] is equiripple except for one peak which is larger than the rest. The position and width of this peak can be set independently. The position of the peak is given by[7]

${\displaystyle x_{\text{max}}=1-2\operatorname {sn} ^{2}(v|\kappa )+2{\dfrac {\operatorname {sn} (v|\kappa )\operatorname {cn} (v|\kappa )}{\operatorname {dn} (v|\kappa )}}Z(v|\kappa )}$

where

${\displaystyle \operatorname {cn} (\varphi |\kappa )}$ is the Jacobi elliptic cosine

${\displaystyle \operatorname {dn} (\varphi |\kappa )}$ is the Jacobi delta amplitude

${\displaystyle Z(\varphi |\kappa )}$ is the Jacobi zeta function

${\displaystyle v}$ is as defined above.

The height of the peak is given by[8]

${\displaystyle Z_{n}(x_{\text{max}}|\kappa )=\cosh 2n{\bigl (}\sigma _{\text{max}}Z(v|\kappa )-\varPi (\sigma _{\text{max}},v|\kappa ){\bigr )}}$

where

${\displaystyle \varPi (\phi _{1},\phi _{2}|\kappa )}$ is the incomplete elliptic integral of the third kind

${\displaystyle \sigma _{\text{max}}=F\left(\left.\sin ^{-1}\left({\dfrac {1}{\kappa \operatorname {sn} (v|\kappa )}}{\sqrt {\dfrac {x_{\text{max}}-x_{\mathrm {L} }}{x_{\text{max}}+1}}}\right)\right|\kappa \right)}$

${\displaystyle x_{\mathrm {L} }}$ is the position on the left limb of the peak which is the same height as the equiripple peaks.

The Jacobi eta function can be defined in terms of a Jacobi auxiliary theta function,[9]

${\displaystyle H(\varphi |\kappa )=\theta _{1}(a|b)}$

where,

${\displaystyle a={\frac {\pi \varphi }{2K'(\kappa )}}}$

${\displaystyle b=\exp \left(-{\frac {\pi K'(\kappa )}{K(\kappa )}}\right)}$

${\displaystyle K'(\kappa )=K({\sqrt {1-\kappa ^{2}}})\ .}$[10]