In mathematics, Zolotarev polynomials are polynomials used in approximation theory. They are sometimes used as an alternative to the Chebyshev polynomials where accuracy of approximation near the origin is of less importance. Zolotarev polynomials differ from the Chebyshev polynomials in that two of the coefficients are fixed in advance rather than allowed to take on any value. The Chebyshev polynomials of the first kind are a special case of Zolotarev polynomials. These polynomials were introduced by Russian mathematician Yegor Ivanovich Zolotarev in 1868.
Zolotarev polynomials of degree
n
x
Zn(x,\sigma)=xn-\sigmaxn-1+ … +akxk+ … +a0 ,
where
\sigma
an-1
ak\inR
Zn(x)
[-1,1]
A subset of Zolotarev polynomials can be expressed in terms of Chebyshev polynomials of the first kind,
Tn(x)
0\le\sigma\le\dfrac{1}{n}\tan2\dfrac{\pi}{2n}
then
Zn(x,\sigma)=(1+\sigma)nTn\left(
| x-\sigma | |
| 1+\sigma |
\right) .
For values of
\sigma
\sigma=0
\sigma
Zn(x,-\sigma)=(-1)nZn(-x,\sigma) .
The Zolotarev polynomial can be expanded into a sum of Chebyshev polynomials using the relationship[3]
Zn(x)=
| n | |
| \sum | |
| k=0 |
akTk(x) .
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 (
q
[-1,1]
p
n=p+q
p=q
n
Zn(x|\kappa)=
| (-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
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)
v=\dfrac{p}{n}K(\kappa)
H(\varphi)
F(\varphi|\kappa)
K(\kappa)
K(\kappa)=F\left(\left.
| \pi | |
| 2 |
\right|\kappa\right)
\kappa
\operatorname{sn}(\varphi|\kappa)
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[6]
xmax=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
\operatorname{cn}(\varphi|\kappa)
\operatorname{dn}(\varphi|\kappa)
Z(\varphi|\kappa)
v
The height of the peak is given by[7]
Zn(xmax|\kappa)=\cosh2nl(\sigmamaxZ(v|\kappa)-\varPi(\sigmamax,v|\kappa)r)
where
\varPi(\phi1,\phi2|\kappa)
\sigmamax=F\left(\left.\sin-1\left(\dfrac{1}{\kappa\operatorname{sn}(v|\kappa)}\sqrt\dfrac{xmax-xL}{xmax+1}\right)\right|\kappa\right)
xL
The Jacobi eta function can be defined in terms of a Jacobi auxiliary theta function,[8]
H(\varphi|\kappa)=\theta1(a|b)
where,
a=
| \pi\varphi | |
| 2K'(\kappa) |
b=\exp\left(-
| \piK'(\kappa) | |
| K(\kappa) |
\right)
K'(\kappa)=K(\sqrt{1-\kappa2}) .
The polynomials were introduced by Yegor Ivanovich Zolotarev in 1868 as a means of uniformly approximating polynomials of degree
xn+1
xn+1
n
2-n
xn+1-\sigmaxn
n-1
2-n\left(\dfrac{1+\sigma}{1+n}\right)n+1 .
The procedure was further developed by Naum Achieser in 1956.[11]
Zolotarev polynomials are used in the design of Achieser-Zolotarev filters. They were first used in this role in 1970 by Ralph Levy in the design of microwave waveguide filters.[12] Achieser-Zolotarev filters are similar to Chebyshev filters in that they have an equal ripple attenuation through the passband, except that the attenuation exceeds the preset ripple for the peak closest to the origin.[13]
Zolotarev polynomials can be used to synthesise the radiation patterns of linear antenna arrays, first suggested by D.A. McNamara in 1985. The work was based on the filter application with beam angle used as the variable instead of frequency. The Zolotarev beam pattern has equal-level sidelobes.[14]
xn