Zolotarev polynomials
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.
Definition and properties
Zolotarev polynomials of degree
where
A subset of Zolotarev polynomials can be expressed in terms of Chebyshev polynomials of the first kind,
then
For values of
The Zolotarev polynomial can be expanded into a sum of Chebyshev polynomials using the relationship[3]
In terms of Jacobi elliptic functions
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 (
- where
is the Jacobi eta function is the incomplete elliptic integral of the first kind is the quarter-wave complete elliptic integral of the first kind. That is, [6] is the Jacobi elliptic modulus 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]
- where
is the Jacobi elliptic cosine is the Jacobi delta amplitude is the Jacobi zeta function is as defined above.
The height of the peak is given by[8]
- where
is the incomplete elliptic integral of the third kind is the position on the left limb of the peak which is the same height as the equiripple peaks.
Jacobi eta function
The Jacobi eta function can be defined in terms of a Jacobi auxiliary theta function,[9]
- where,
[10]
Applications
The polynomials were introduced by Yegor Ivanovich Zolotarev in 1868 as a means of uniformly approximating polynomials of degree
The procedure was further developed by Naum Achieser in 1956.[12]
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.[13] 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.[14]
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.[15]
References
- ↑ Pinkus, pp. 463–464
- ↑ Pinkus, p. 464
- ↑ Zahradnik & Vlček, p. 58
- ↑ Cameron et al., p. 400
- ↑ Zahradnik & Miroslav, pp. 57–58
- ↑ Beebe, p. 624
- ↑ Zahradnik & Miroslav, p. 58
- ↑ Zahradnik & Miroslav, p. 58
- ↑ Beebe, p. 679
- ↑ Beebe, p. 625
- ↑ Newman & Reddy, p. 310
- ↑ Newman & Reddy, pp. 310, 316
- ↑ Hansen, p.87
- ↑ Cameron et al., p. 399
- ↑ Hansen, p.87
Bibliography
- Achieser, Naum, Hymnan, C.J. (trans), Theory of Approximation, New York: Frederick Ungar Publishing, 1956. Dover reprint 2013 ISBN:0486495434.
- Beebe, Nelson H.F., The Mathematical-Function Computation Handbook, Springer, 2017 ISBN:978-3-319-64110-2.
- Cameron, Richard J.; Kudsia, Chandra M.; Mansour, Raafat R., Microwave Filters for Communication Systems, John Wiley & Sons, 2018 ISBN:1118274342.
- Hansen, Robert C., Phased Array Antennas, Wiley, 2009 ISBN:0470529172.
- McNamara, D.A., "Optimum monopulse linear array excitations using Zolotarev Polynomials", Electron, vol. 21, iss. 16, pp. 681–682, August 1985.
- Newman, D.J., Reddy, A.R., "Rational approximations to
II", Canadian Journal of Mathematics, vol. 32, no. 2, pp. 310–316, April 1980. - Pinkus, Allan, "Zolotarev polynomials", in, Hazewinkel, Michiel (ed), Encyclopaedia of Mathematics, Supplement III, Springer Science & Business Media, 2001 ISBN:1402001983.
- Vlček, Miroslav, Unbehauen, Rolf, "Zolotarev polynomials and optimal FIR filters", IEEE Transactions on Signal Processing, vol. 47, iss. 3, pp. 717–730, March 1999 (corrections July 2000).
- Zahradnik, Pavel; Vlček, Miroslav, "Analytical design of 2-D narrow bandstop FIR filters", pp. 56–63 in, Computational Science — ICCS 2004: Proceedings of the 4th International Conference, Bubak, Marian; van Albada, Geert D.; Sloot, Peter M.A.; Dongarra, Jack (eds), Springer Science & Business Media, 2004 ISBN:3540221298.
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