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Projective view at optimization problem for multiband filter. (English) Zbl 07352300
Summary: The best uniform rational approximation of the sign function on two intervals separated by zero was explicitly found by E.I. Zolotarëv in 1877. This optimization problem is the initial step in the staircase of the so called approximation problems for multiband filters which are of great importance for electrical engineering. We show that known in the literature optimality criterion for this problem may be contradictory since it does not take into account the projective invariance of the problem. We propose a new consistently projective formulation of this problem and give a constructive optimality criterion for it.
41A20 Approximation by rational functions
41A50 Best approximation, Chebyshev systems
49K35 Optimality conditions for minimax problems
94Cxx Circuits, networks
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[1] Zol E. I. Zolotar\"ev, Application of Elliptic Functions to Questions on Functions Deviating Least and Most from Zero, Zap. Imp. Akad. Nauk St. Petersburg, vol. 30, no. 5 (1877), pp. 1-71.
[2] Akh N. I. Ach\"eser, Sur un probl\`eme de E. Zolotar\"ev, Bull. Acad. Sci. de l’URSS, VII s\'er., 1929, no. 10, pp. 919\nobreakdash –931.
[3] Akhiezer, N. I., Elements of the theory of elliptic functions, Translations of Mathematical Monographs 79, viii+237 pp. (1990), American Mathematical Society, Providence, RI · Zbl 0694.33001
[4] Achieser, N. I., Theory of approximation, x+307 pp. (1992), Dover Publications, Inc., New York
[5] Sti E. Stiefel, Le probl\`eme d’approximation dans la th\'eorie des filtres \'electriques, Colloque sur l’analyse num\'erique \`a Mons, 1961, pp. 81-87.
[6] Stiefel, E., Methods-old and new-for solving the Tchebycheff approximation problem, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 1, 164-176 (1964) · Zbl 0141.33502
[7] Amer, R. A.-R.; Schwarz, H. R., Contributions to the approximation problem of electrical filters, Mitt. Inst. Angew. Math. Z\"{u}rich, 9, 99 pp. (1964) · Zbl 0125.07601
[8] Gon\v{c}ar, A. A., The problems of E. I. Zolotarev which are connected with rational functions, Mat. Sb. (N.S.), 78 (120), 640-654 (1969)
[9] Maloz\"{e}mov, V. N., The synthesis problem for a multipole electrical filter, Zh. Vychisl. Mat. i Mat. Fiz., 19, 3, 601-609, 796 (1979) · Zbl 0414.90092
[10] Cauer, Wilhelm, Theorie der linearen Wechselstromschaltungen, xxiv+769 pp. (1954), Akademie-Verlag, Berlin · Zbl 0100.39603
[11] Bogatyrev, Andrei, Extremal polynomials and Riemann surfaces, Springer Monographs in Mathematics, xxvi+150 pp. (2012), Springer, Heidelberg · Zbl 1252.30001
[12] Bogatyr\"{e}v, A. B., Chebyshev representation of rational functions, Mat. Sb.. Sb. Math., 201 201, 11-12, 1579-1598 (2010) · Zbl 1213.41007
[13] Bogatyrev, A. B.; Gore\u{\i}nov, S. A.; Lyamaev, S. Yu., Analytical approach to multiband filter synthesis and comparison to other approaches, Problemy Peredachi Informatsii. Probl. Inf. Transm., 53 53, 3, 260-273 (2017) · Zbl 1387.94027
[14] Remez, E. Ja., Osnovy chislennykh metodov chebyshevskogo priblizheniya, 624 pp. (errata insert) pp. (1969), “Naukova Dumka”, Kiev
[15] Veidinger, L., On the numerical determination of the best approximations in the Chebyshev sense, Numer. Math., 2, 99-105 (1960) · Zbl 0090.33702
[16] Fuchs, W. H. J., On Chebyshev approximation on sets with several components. Aspects of contemporary complex analysis, Proc. NATO Adv. Study Inst., Univ. Durham, Durham, 1979, 399-408 (1980), Academic Press, London-New York
[17] Kelley, John L., General topology, Graduate Texts in Mathematics, xiv+298 pp. (1975), Springer-Verlag, New York-Berlin · Zbl 1358.54001
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