Least Deviation Problems

  • Andrei Bogatyrev
Part of the Springer Monographs in Mathematics book series (SMM)


We begin this chapter by listing areas of science and technology where we come across problems relating to optimization of the uniform norm. After that we investigate least deviation problems using methods of convex analysis. We deduce a generalized alternation principle which completely characterizes solutions of such problems. In giving the definition of an extremal polynomial in the introduction we were motivated by this principle.


Extremal Point Real Axis Uniform Norm Affine Plane Deviation Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Abdulle, A.: On roots and error constants of optimal stability polynomials. BIT 42, 177-182 (2000)Google Scholar
  2. 2.
    Abel, N.H.: Sur l’intégration la formule diférentielle \( \frac {pdx}{\sqrt{R}}\), R et ρ etant des fonctions entieres. J. Reine Angew. Math. 1, 105-144 (1826).Google Scholar
  3. 3.
    Ahlfors, L.V.: Lectures on Quasiconformal Mappings, Van Nostrand: Toronto, Ont. (1966)Google Scholar
  4. 4.
    Ahlfors, L., Bers, L: Spaces of Riemann Surfaces and Quasiconformal Mappings, Moscow, Inostrannaya Literatura (1961). (A Russian translation of several papers: Bers, L.: Quasiconformal mappings and Teichmuller’s theorem. In: Analytic Functions, pp. 89–119, Princeton Univ. Press, Princeton, N.J. (1960); Ahlfors, L.: The complex analytic structure of the space of closed Riemann surfaces. Ibid., pp. 45–66; Bers, L.: Spaces of Riemann surfaces. In: Proc. Internat. Congr. Math. (Edinburgh, 1958), pp. 349–361; Bers, L.: Simultaneous uniformization. Bull. Amer. Math. Soc. 66, 94–97 (1960); Bers, L.: Holomorphic differentials as functions of moduli. Bull. Amer. Math. Soc. 67, 206–210 (1961); Ahlfors, L. On quasiconformal mappings. J. Anal. Math. 3 1–58; correction, 207–208 (1954))Google Scholar
  5. 5.
    Ahlfors, L.V.: The complex analytic structure of the space of closed Riemann surfaces. In: Analytic Functions, pp. 45–66, Princeton Univ. Press, Princeton, N.J. (1960)Google Scholar
  6. 6.
    Akaza, T., Inoue, K.: Limit sets of geometrically finite free kleinian groups. Tohoku Math. J. 36 1–16 (1984)Google Scholar
  7. 7.
    Akaza, T.: Singular sets of some Kleinian groups. Nagoya Math. J. 29, 145-162 (1967)Google Scholar
  8. 8.
    Akaza, T., Inoue, K.: On the limit set of a geometrically finite Kleinian group Sci. Rep. Kanazawa Univ. 27, 85-115 (1982)Google Scholar
  9. 9.
    Akhiezer, N.I.: Über einige Funktionen die in gegebenen Intervallen am wenigsten von Null abweichen. Izv. Kazan Fiz.-Mat. Obshch. textbf3, no. 3 (1928)Google Scholar
  10. 10.
    Akhiezer, N.I.: Über einige functionen welche in zwei gegebenen Intervellen am wenigsten von Null abweichen I–III. Izv. Akad. Nauk SSSR 9, 1163–1202 (1932); 3, 309–344 (1933); 4, 499–536 (1933)Google Scholar
  11. 11.
    Akhiezer, N.I.: Orthogonal polynomials on several intervals. Dokl. Akad. Nauk SSSR. 134:1, 9–12 (1960). (in Russian)Google Scholar
  12. 12.
    Akhiezer, N.I.: Lectures in the Theory of Approximation. Nauka, Moscow (1965), 2nd edn. English transl. of 1st edn.: Achieser, N.I.: Theory of Approximation. Dover Publ., New York (1992)Google Scholar
  13. 13.
    Akhiezer, N.I., Levin B.Ya.: Inequalities for derivatives similar to Bernoulli’s inequality. Dokl. Akad. Nauk SSSR 117:5, 735–738 (1957). (in Russian)Google Scholar
  14. 14.
    Antoniou, A.: Digital Filters: Analysis, Design, and Applications, McGraw-Hill (1979)Google Scholar
  15. 15.
    Arnold, V.I.: Arnold’s Problems. Phasis, Moscow (2000). English transl.: Springer-Verlag, Berlin; PHASIS, Moscow (2004)Google Scholar
  16. 16.
    Arbarello, E., Cornalba, M., Grifiths, P.A., Harris, J.: Geometry of Algebraic Curves I-II. Springer, New York 1985.Google Scholar
  17. 17.
    Artin, E.: Theorie der Zöpfe. Abh. Math. Semin. Univ. Hamburg. 4, 47-72 (1925)Google Scholar
  18. 18.
    Baker, H.F.: Abel Theorem and Allied Theory. Cambridge (1897)Google Scholar
  19. 19.
    Belokolos, E.D., Enolskii, V.Z.: Reduction of abelian unctions and completely integrable equations. J. Math. Sci. 106 3395–3486 (2001); 108 295–374 (2002)Google Scholar
  20. 20.
    Bers, L. Uniformization, moduli and Kleinian groups. Bull. London Math. Soc. 4, 257-300 (1972)Google Scholar
  21. 21.
    Birman, J.S.: Braids, Links, and Mapping Class Groups. Princeton University Press, Princeton, NJ: (1974)Google Scholar
  22. 22.
    Bobenko, A.I., Klein, Ch. (Eds.): Computational Approach to Riemann Surfaces. Lecture Notes in Mathematics, Vol. 2013, Springer (2011)Google Scholar
  23. 23.
    Bogatyrev, A.B.: Chebyshev polynomials and navigation in moduli space of hyperelliptic curves. Russian J. Numer. Anal. Math. Modelling. 14:3, 205–220 (1999)Google Scholar
  24. 24.
    Bogatyrev, A.B.: Fibers of periods map are cells? J. Comput. Appl. Math. 153, 647-548 (2003)Google Scholar
  25. 25.
    Bogatyrev, A.B.: Effectve computation of optimal stability polynomials. Calcolo. 41:4, 247-156 (2004)Google Scholar
  26. 26.
    Bogatyrev. A.B.: Chebyshev representation for rational functions. Sb. Math. 201 1579 - 1598 (2010)Google Scholar
  27. 27.
    Bogatyrev, A.B.: Computations in moduli spaces. Comp. Methods and Function Theory 7:1 (2007)Google Scholar
  28. 28.
    Burau, W.: Über Zopfinvarianten. Abh. Math. Semin. Univ. Hamburg. 4. 47-72 (1925)Google Scholar
  29. 29.
    Burnside, W.: On a class of authomorphic functions. Proc. London Math. Soc. 23, 49-88 (1892)Google Scholar
  30. 30.
    Buser, P., and Seppala, M.: Computing on Riemann Surfaces. Topology and Teichmuller spaces (Katinkulta, 1995), 5-30, World Sci. Publ., River Edge, NJ, 1996Google Scholar
  31. 31.
    Bakhvalov, N. S., Zhidkov, N. P., Kobel’kov, G. M.: Numerical Methods. Moscow, Nauka (1987). (in Russian)Google Scholar
  32. 32.
    Beardon, A.F.: The Geometry of Discrete Groups. Graduate Texts in Mathematics 91. Springer-Verlag, New York (1995)Google Scholar
  33. 33.
    Bernstein, S.N.: Extremal Properties of Polynomials. Moscow, ONTI (1937).Google Scholar
  34. 34.
    Bers, L.: Holomorphic differentials as functions of moduli. Bull. Amer. Math. Soc. 67, 206–210 (1961).Google Scholar
  35. 35.
    Bobenko, A. I. Schottky uniformization and finite-gap integration. Dokl. Akad. Nauk SSSR 295, 268–272 (1987). English transl. in Sov. Math., Dokl. 36:1, 38-42 (1987) [See also Chap. 5 in: Belokolos, E. D., Bobenko, A.I., Enol’skii, V.Z., Its, A.R., Matveev, V.B.: Algebro-Geometric Approach to Nonlinear Integrable Equations. Springer, Berlin (1994)]Google Scholar
  36. 36.
    Bogatyrev, A.B.: A geometric method for solving a series of integral Poincaré-Steklov equations. Mat. Zametki 63:3, 343–353 (1998). English transl. in Math. Notes 63, 302-310 (1998)Google Scholar
  37. 37.
    Bogatyrev, A.B.: Effective computation of Chebyshev polynomials for several intervals. Mat. Sb. 190:11, 15–50 (1999). English transl. in Sb. Math. 190, 1571–1605 (1999).Google Scholar
  38. 38.
    Bogatyrev, A.B.: Manifolds of support sets of Chebyshev polynomials. Mat. Zametki 67, 828–836 (2000). English transl. in Math. Notes 67, 699–706 (2000)Google Scholar
  39. 39.
    Bogatyrev, A.B.: Poincare–Steklov integral equations and the Riemann monodromy problem. Funktsional. Anal. i Prilozhen. 34:2, 9–22 (2000). English transl. in Funct. Anal. Appl. 34, 86-97 (2000)Google Scholar
  40. 40.
    Bogatyrev, A.B.: Effective approach to least deviation problems. Mat. Sb. 193:12, 21–41 (2002). English transl. in Sb. Math. 193, 1749–1769 (2002)Google Scholar
  41. 41.
    Bogatyrev, A.B.: Representation of moduli spaces of curves and the computation of extremal polynomials. Mat. Sb. 194:4, 3–28 (2003). English transl. in Sb. Math. 194, 469–494 (2003)Google Scholar
  42. 42.
    Bogatyrev, A.B.: A combinatorial description of a moduli space of curves and of extremal polynomials. Mat. Sb. 194:10, 27–48 (2003). English transl. in Sb. Math. 194, 1451–1473 (2003).Google Scholar
  43. 43.
    Bogatyrev, A.B.: Effective solution of the problem of the optimal stability polynomial. Mat. Sb. 196:7, 27–50 (2005). English transl. in Sb. Math. 196, 959–981 (2005)Google Scholar
  44. 44.
    Cauer, W.: Ein Interpolationsproblem mit Functionen mit positivem Realteil. Math. Z. 38 (1934)Google Scholar
  45. 45.
    Cauer, W.: Theorie der Linearen Wechselstromschaltungen. Akademie Verlag, Berlin (1954)Google Scholar
  46. 46.
    Chebotarev, N.G.: The Theory of Algebraic Functions. OGIZ, Moscow-Leningrad (1948). (in Russian)Google Scholar
  47. 47.
    Chebyshev, P.L.: Théorie des mécanismes connus sous le nom de parallélogrammes. Mém. Acad. Sci. Pétersb. 7, 539-568 (1854).Google Scholar
  48. 48.
    Chekhov, L.O.: Matrix models: Geometry of moduli spaces and exact solutions. Teoret. Mat. Fiz. 127, 179-252 (2001). English transl. in Theor. Math. Phys. 127, 557-618 (2001)Google Scholar
  49. 49.
    Chen, X., Parks, T.W.: Analytic design of optimal FIR narrow-band filters using Zolotarev polynomials. IEEE Trans. on Circuits and Systems. 33, 1065-1071 (1986)Google Scholar
  50. 50.
    Crowdy, D.G., Marshall, J.S.: Conformal mappings between canonical multiply connected domains. Computational Methods and Function Theory, 6 59-76 (2006)Google Scholar
  51. 51.
    Crowdy, D.G.: Schwarz–Christoffel mappings to unbounded multiply connected polygonal regions. Math. Proc. Cambridge Philos. Soc., 142 319–339 (2007)Google Scholar
  52. 52.
    Crowdy, D.G., Marshall, J.S.: Green’s functions for Laplace’s equation in multiply connected domains. IMA. J. Appl. Math., 72 278–301 (2007)Google Scholar
  53. 53.
    Deconinck, B., van Hoeij, M.: Computing Riemann matrices of algebraic curves. Physica D 152 28-46 (2001)Google Scholar
  54. 54.
    Deconinck, B., Heil, M., Bobenko, A., van Hoeij, M., Schmies, M.: Computing Riemann theta functions. Math. Comp. 73 1417-1442 (2004)Google Scholar
  55. 55.
    Deconinck, B., Patterson, M.: Computing the Abel map. Physica D 237 3214-3232 (2008)Google Scholar
  56. 56.
    Dubrovin, B.A.: Theta functions and non-linear equations. Uspekhi Mat. Nauk 36:2, 11–80 (1981). English transl. in Russian Math. Surveys 36:2, 11–92 (1981)Google Scholar
  57. 57.
    Earl, C.J.: On variation of projective structures. In: Riemann Surfaces and Related Topics, pp. 87-99. Princeton University Press, Princeton, NJ (1980)Google Scholar
  58. 58.
    Encyclopaedia of Mathematics. Sovetskaya Entsiklopediya, Moscow (1984). English transl. Vols. 1-5, Kluwer, Dordrecht (1988-1994)Google Scholar
  59. 59.
    Enol’skii, V.Z., Kostov, N.A.: On the geometry of elliptic solitons. Acta Appl. Math. 36, 57-86 (1994)Google Scholar
  60. 60.
    Farkas, H., Kra, I.: Riemann Surfaces. Springer, New York (1992)Google Scholar
  61. 61.
    Fay, J.: Theta Functions on Riemann Surfaces. Lecture Notes in Mathematics, Vol. 352. Springer, Berlin-New York (1973)Google Scholar
  62. 62.
    Fock, V.V., Dual Teichmüller spaces. ArXiv: dg-ga/9702018, 1998.Google Scholar
  63. 63.
    Fomenko, A.T., Fuchs, D.B.: A Course of Homotopic Topology. Nauka, Moscow (1989). (in Russian)Google Scholar
  64. 64.
    Ford, L.R.: Automorphic Functions. McGraw-Hill, New York (1929)Google Scholar
  65. 65.
    Franklin, J.N.: Numerical stability in digital and analog computation for diffusion problems. J. Math. Phys. 37, 305–315 (1959)Google Scholar
  66. 66.
    Gardiner, F.P.: Teichmüller Theory and Quadratic Differentials. Wiley, New York (1987)Google Scholar
  67. 67.
    Gesztesy, F., Weikard. R.: Picard potentials and Hill’s equation on a torus. Acta Math. 176, 73–107 (1996)Google Scholar
  68. 68.
    Gesztesy, F., Weikard. R.: Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies—an analitic approach. Bull. Amer. Math. Soc. 35 271-317 (1998)Google Scholar
  69. 69.
    Griffiths, Ph., Harris, J.: Principles of Algebraic Geometry. Wiley, New York (1994)Google Scholar
  70. 70.
    Gunning, R.C.: Lectures on Riemann Surfaces. Princeton University Press, Princeton, NJ (1966)Google Scholar
  71. 71.
    Hairer, E., Wanner, G. Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin (1996)Google Scholar
  72. 72.
    Hejhal, D.A.: Sur les parameters accessoires pour l’uniformization de Schottky. C. R. Acad. Sci. Paris Sér. A. 279, 713-716 (1974)Google Scholar
  73. 73.
    Hejhal, D.A.: On Schottky and Teichmüller spaces. Adv. Math. 15 133-156 (1975)Google Scholar
  74. 74.
    Hejhal, D.A.: The variational theory of linearly polymorphic functions. J. Anal. Math. 30, 215-264 (1976)Google Scholar
  75. 75.
    van Hoeij, M.: An algorithm for computing the Weierstrass normal form. ISSAC ’95 Proceedings, 90-95 (1995).Google Scholar
  76. 76.
    van Hoeij, M.: Computing parametrizations of rational algebraic curves. ISSAC ’94 Proceedings, 187-190 (1994)Google Scholar
  77. 77.
    van der Houwen, P.J., Kok, J.: Numerical solution of a maximal problem. Report 124/71 TW, Mathematical Centre, Amsterdam (1971)Google Scholar
  78. 78.
    Hurwitz, A.: Über Riemannsche Flächen mit gegebenen Verzweigungspunkten. Math. Ann. 39, 1–61 (1891)Google Scholar
  79. 79.
    Hurwitz, A.: Vorlesungen über Allgemeine Funktionentheorie und Elliptische Funktione. Herausgegeben und erganzt durch einen Abschnitt uber geometrische Funktionentheorie von R. Courant. Springer-Verlag, Berlin-New York (1964)Google Scholar
  80. 80.
    Hubbard, J.H.: On monodromy of projective structures. In: Riemann Surfaces and Related Topics, pp. 257–275. Princeton University Press, Princeton, NJ (1980)Google Scholar
  81. 81.
    Hubbard, J., Masur, H.: Quadratic differentials and foliations. Acta Math. 142:1 (1979), 221–274.Google Scholar
  82. 82.
    Igusa, J.: Problems on Abelian functions at the time of Poincare and some at present. Bull. Amer. Math. Soc., 6 161-174 (1982)Google Scholar
  83. 83.
    Klein, F.: Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. I. J.Springer, Berlin (1926)Google Scholar
  84. 84.
    Kontsevich, M.L.: Intersection theory on the moduli space of curves. Funktsional. Anal. i Prilozhen. 25:2, 50–57 (1991). English transl. in Funct. Anal. Appl. 25:2, 123–129 (1991)Google Scholar
  85. 85.
    Kra, I.: Automorphic Forms and Kleinian Groups. Mathematics Lecture Note Series., W. A. Benjamin, Inc., Reading, Mass. (1972)Google Scholar
  86. 86.
    Kreǐn, M.G, Levin, B.Ya., Nudel’man, A.A.: On special representations of polynomials that are positive on a system of closed intervals, and some applications. In: Functional Analysis, Optimization, and Mathematical Economics, pp. 56–114, Univ. Press, New York, Oxford (1990)Google Scholar
  87. 87.
    Kreǐnes, E.M.: Rational Functions with Few Critical Values. Ph.D. Thesis, Moscow State University (2001). (in Russian).Google Scholar
  88. 88.
    Krichever, I.M.: Integration of nonlinear equations by the methods of algebraic geometry. Funktsional. Anal. i Prilozhen. 11:2, 15–32 (1977). English transl. in Funct. Anal. Appl. 11:2, 12–26 (1977)Google Scholar
  89. 89.
    Krushkal’, S.L., Apanasov, B.N., Gusevskii, N.A.: Kleinian Groups and Uniformization in Examples and Problems. Nauka, Novosibirsk (1981). English transl. American Mathematical Society, Providence, RI (1986).Google Scholar
  90. 90.
    Lando, S.K., Ramified coverings of the two-dimensional sphere and intersection theory in spaces of meromorphic functions on algebraic curves. Uspekhi. Mat. Nauk. 57:3, 29–98 (2002). English transl. in Russian Math. Surveys 57, 463–533 (2002)Google Scholar
  91. 91.
    Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and their Applications. Encyclopaedia of Mathematical Sciences, Vol. 141. Berlin: Springer, New York-Berlin (2004)Google Scholar
  92. 92.
    Lavrent’ev, M.A., Shabat, B.V.: Methods of the Theory of Functions of a Complex Variable. Nauka, Moscow (1965). (in Russian)Google Scholar
  93. 93.
    Lebedev, V.I.: How rigid systems of differential equations can be solved by explicit methods. In: Marchuk, G.I. (Ed.) Vychisltel’nye Protsessy i Sistemy, Vol 8, pp. 237–292. Nauka, Moscow (1991). (in Russian)Google Scholar
  94. 94.
    Lebedev, V.I., Medovikov, A.A.: An explicit method of the second order of accuracy for solving stiff systems of ordinary differential equations. Izv. Vyssh. Uchebn. Zaved. Mat. 10:9, 37–52 (1995). English transl. in Russ. Math. 42:9, 52-60 (1998)Google Scholar
  95. 95.
    Lebedev, V.I.: A new method for determining the roots of polynomials of least deviation on a segment with weight and subject to additional conditions I, II. Russian J. Numer. Anal. Math. Modelling. 8:3, 195-222; 8:5, 397-426 (1993)Google Scholar
  96. 96.
    Lebedev, V.I.: Zolotarev polynomials and extremum problems Russian J. Numer. Anal. Math. Modelling. 9:3, 231-263 (1994)Google Scholar
  97. 97.
    Lebedev, V.I.: Extremal polynomials with restrictions and optimal algorithms. In: Advanced Mathematics: Computation and Applications (Eds. Alekseev, A.S., Bakhvalov, N.S.), pp. 491-502. NCC Publishers, Novosibirsk (1995)Google Scholar
  98. 98.
    Lomax, H.: On the construction of highly stable, explicit numerical methods for integrating coupled ordinary differential equations with parasitic eigenvalues. NASA Technical Note, NASATND/4547 (1968)Google Scholar
  99. 99.
    Malyshev, V.A.: The Abel equation. Algebra i Analiz, 13:6, 1–55 (2001). English transl. in St. Petersburg Math. J. 13, 893-938 (2002).Google Scholar
  100. 100.
    Markov, A.A.: Lectures on functions deviating least from zero. In: Markov, A.A.: Selected Papers on Continued Fractions and Functions Least Deviating from Zero. Gostekhizdat, Mosow-Leningrad (1948). (in Russian)Google Scholar
  101. 101.
    Markov, A.A.: A question put by D.I.Mendeleev. Izv. Pererburg. Akad. Nauk, issue 62, 1-24 (1889). (in Russian)Google Scholar
  102. 102.
    Markov V.A.: Functions Deviating Least from Zero on a Prescribed Interval. St.-Petersburg (1892). (in Russian)Google Scholar
  103. 103.
    Medovikov, A.A.: High order explicit methods for parabolic equations. BIT 38, 372-390 (1998)Google Scholar
  104. 104.
    Meiman, N.N.: On the theory of polynomials deviating least from zero. Dokl. Akad. Nauk SSSR 130:2, 257-260 (1960). English transl. in Sov. Math., Dokl. 1, 41-44 (1960)Google Scholar
  105. 105.
    Metzger, C.: Métodes de Runge–Kutta de Rang Superieur á l’Ordre. Thesis, Univ. Grenoble (1967)Google Scholar
  106. 106.
    Mityushev, V.V.: Convergence of the Poincaré series for the classical Schottky groups. Proc. Amer. Math. Soc., 126, 2399-2406 (1998)Google Scholar
  107. 107.
    Mumford, D.: Curves and Their Jacobians. University of Michigan Press, Ann Arbor (1975)Google Scholar
  108. 108.
    Mumford, D.: Tata Lectures on Theta, I. Birkhäuser, Boston-Basel (1983)Google Scholar
  109. 109.
    Mumford, D., Series, C., Wright, D.: Indra’s Pearls: The Vision of Felix Klein, Cambridge University Press (2002), 416 p.Google Scholar
  110. 110.
    Myrberg, P.J.: Zur Theorie der Convergenz der Poincareschen Reihen. Ann. Acad. Sci. Fenn. (A). 9:4, 1–75 (1916)Google Scholar
  111. 111.
    Natanzon, S.M.: Moduli of Riemann Surfaces, Real Algebraic Curves, and Their Superanalogs. Moscow Centre for Continued Mathematical Education, Moscow (2003). English transl. American Mathematical Society, Providence, RI (2004)Google Scholar
  112. 112.
    Pakovich, F.B.: Elliptic polynomials. Uspekhi Mat. Nauk 50:6, 203-204. English transl. in Russian Math. Surveys 50, 1292-1294 (1995)Google Scholar
  113. 113.
    Pakovitch, F.: Combinatoire des arbes planaires et arithmétique des courbes hyperelliptiques Ann. Inst. Fourier, 48, 323-351 (1998)Google Scholar
  114. 114.
    Paszkowski, S.: Numerical Applications of Chebyshev Polynomials and Series. Panstwowe Wydawnictwo Naukowe, Warszawa (1983) (in Polish)Google Scholar
  115. 115.
    Peherstorfer, F.: On Bernstein-Szegö orthogonal polynomials on several intervals. II. Orthogonal polynomials with periodic recurrence coefficients. J. Approx. Theory. 64, 123-161 (1991)Google Scholar
  116. 116.
    Peherstorfer, F.: Orthogonal and extremal polynomials on several intervals. J. Comput. Appl. Math. 48, 187-205 (1993)Google Scholar
  117. 117.
    Peherstorfer, F.: Deformation of minimal polynomials and approximation of several intervals by an inverse polynomial mapping. J. Approx. Theory. 111, 180-195 (2001)Google Scholar
  118. 118.
    Peherstorfer, F., Schiefermayr, K.: Description of extremal polynomials on several intervals and their computation I, II. Acta Math. Hungar. 83, 71-102, 103-128 (1999)Google Scholar
  119. 119.
    Poincaré, H.: Sur la réduction des intégrales Abéliennes. Bull. Soc. Math. France, 12 124-143 (1884)Google Scholar
  120. 120.
    Poincaré, A.: Théorie des groupes fuchsiennes Acta Math. 1, 1-62 (1882)Google Scholar
  121. 121.
    Poincaré, A.: Sur les fonctions fuchsiennes. Acta Math. 1, 193-294 (1882)Google Scholar
  122. 122.
    Poincaré, A.: Analyse des travaux scientifiques de Henri Poincaré faite par lui-même. Acta Math. 38, 36-135 (1921)Google Scholar
  123. 123.
    Prasolov, V.V., Schwartsman, O.V.: Alphabeth of Riemann Surfaces, Phasis: Moscow (1999). (in Russian)Google Scholar
  124. 124.
    Rauch, H.E.: Weierstrass points, branch points, and moduli of Riemann surfaces. Comm. Pure Appl. Math. 12 543-560, (1959)Google Scholar
  125. 125.
    Rauch, H.E.: On the transcendential moduli of algebraic Riemann surfaces. Proc. Nat. Acad. Sci. USA. 41, 42-49 (1955)Google Scholar
  126. 126.
    Remez, Ya.I.: General Numerical Methods of Chebyshev Approximation. Publishing House of the Academy of Sciences of Ukr.SSR, Kiev (1957). (in Russian)Google Scholar
  127. 127.
    Riha, W.: Optimal stability polynomials. Computing. 9, 37-43 (1972)Google Scholar
  128. 128.
    Robinson, R.: Conjugate algebraic integers in real point sets. Math. Z. 84, 415-427 (1964)Google Scholar
  129. 129.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, NJ (1970)Google Scholar
  130. 130.
    Schiffer, M., Spencer, D.C.: Functionals of finite Riemann surfaces. Princeton University Press, Princeton, NJ (1954)Google Scholar
  131. 131.
    Schmies, M.: Computational Methods for Riemann Surfaces and Helicoids with Handles. PhD thesis, TU Berlin (2005)Google Scholar
  132. 132.
    Schottky, F.: Über eine specielle Function welche bei einer bestimmten linearen Transformation unverändert bleibt. J. Reine Angew. Math. 101, 227-272 (1887)Google Scholar
  133. 133.
    Seppala, M.: Computational methods in the theory of Riemann surfaces and algebraic curves. Proceedings of Workshop on Symbolic and Numeric Computation, (Helsinki, May 30–31, 1991). Computing Centre, University of Helsinki, Research Reports N. 17 145-150 (1991)Google Scholar
  134. 134.
    Seppala, M.: Computation of period matrices of real algebraic curves. Discrete Comput. Geom. 11 65-81 (1994)Google Scholar
  135. 135.
    Shabat, B.V.: Introduction to Complex Analysis. Part II. Nauka, Moscow (1985). English transl. American Mathematical Society, Providence, RI, 1992Google Scholar
  136. 136.
    Shabat, G.B., Zvonkin, A.K.: Plane trees and algebraic numbers. Contemp. Math. 178, 233-275 (1994)Google Scholar
  137. 137.
    Shafarevich, I.R.: Basic Algebraic Geometry, 1-2. Nauka, Moscow (1988). English transl. Springer-Verlag, Berlin (1994)Google Scholar
  138. 138.
    Singer, I.: Best Approximation in Normed Vector Spaces by Elements from Subspaces. Acad. RSR, Bucuresti (1967). (in Romanian)Google Scholar
  139. 139.
    Sodin, M.L., Yuditskii, P.M.: Functions deviating least from zero on closed subsets of the real axis. Algebra i Analiz 4:2, 1–61 (1992). English transl. in St. Petersburg Math. J. 4, 201-249 (1993)Google Scholar
  140. 140.
    Sodin, M.L., Yuditskii, P.M.: Algebraic solution of a problem of E. I. Zolotarev and N. I. Akhiezer on polynomials with smallest deviation from zero. Teor. Funkts., Funkts. Anal. Prilozh. 56, 56-64 (1991). English transl. in J. Math. Sci. (New York) 76, No.4, 2486-2492 (1995)Google Scholar
  141. 141.
    Springer, G.: Introduction to Riemann Surfaces. Addison-Wesley, Reading, Mass. (1957)Google Scholar
  142. 142.
    Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Springer-Verlag, New York-Heidelberg (1973)Google Scholar
  143. 143.
    Strebel, K.: Quadratic Differentials. Springer, Berlin-New York (1984)Google Scholar
  144. 144.
    Suetin, S.P.: Pade approximants and the effective analytic continuation of a power series. Uspekhi Mat. Nauk 57:1, 45-142 (2002). English transl. in Russian Math, Surveys 57, 43-141 (2002)Google Scholar
  145. 145.
    Sullivan, D.: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math., 153, 259-277 (1984)Google Scholar
  146. 146.
    Tikhomirov, V.M.: Some Problems of Approximation Theory, Moscow State University Publishing House, Moscow (1976). (in Russian)Google Scholar
  147. 147.
    Todd, J.: A legacy from Zolotarev. Math Intelligencer. 10:2, 50-53 (1988)Google Scholar
  148. 148.
    Totik, V.: Polynomial inverse images and polynomial inequalities. Acta Math. 187, 139-160 (2001)Google Scholar
  149. 149.
    Tsuji, M.: Potential Theory in Modern Function Theory. Maruzen, Tokyo (1959)Google Scholar
  150. 150.
    Vassiliev, V.A.: Ramified Integrals. Moscow Centre for Continued Mathematical Education, Moscow (2000). English version: Vassiliev, V. A.: Ramified Integrals, Singularities and Lacunas. Mathematics and its Applications, 315, Kluwer, Dordrecht (1995)Google Scholar
  151. 151.
    Vassiliev, V.A.: Introduction to Topology, Phasis, Moscow (1977). English transl. American Mathematical Society, Providence, RI, (2001)Google Scholar
  152. 152.
    Vekua, I.N.: Generalized Analytic Functions. Nauka, Moscow (1988), 2nd edn. English transl. of 1st edn. Pergamon Press, London-Paris-Frankfurt; Addison-Wesley, Reading, Mass. (1962)Google Scholar
  153. 153.
    Verwer, J.W.: Explicit Runge–Kutta methods for parabolic partial differential equations. Appl. Numer. Math. 22, 359-379 (1996)Google Scholar
  154. 154.
    Vlček, M., Unbehauen, R.: Zolotarev Polynomials and optimal FIR filters. IEEE Trans. on Signal Processing. 47, 717-729 (1999)Google Scholar
  155. 155.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, New York (1962)Google Scholar
  156. 156.
    Widom, H.: Extremal polynomials associated with a system of curves in the complex plane. Adv. Math. 3 127-232 (1969)Google Scholar
  157. 157.
    Zakharov, V.E., Zaslavskii, M.M., Kabatchenko, I.M., Matushevskii, G.V., Polnikov, V.G.: Conceptually new wind-wave model. In: The Wind-Driven Air-Sea Interface (Ed. Banner, M.L.), pp. 159-164. The University of New South Wales, Sydney, Australia (1999)Google Scholar
  158. 158.
    Zdravkovska, S.: Topological classification of polynomial maps. Uspekhi Mat. Nauk 25:4, 179–180 (1970). (in Russian)Google Scholar
  159. 159.
    Zieschang, H., Vogt, E., Coldewey, H.-D.: Flachen und Ebene Diskontinuierliche Gruppen. Lecture Notes in Mathematics, Vol. 122, Springer-Verlag, Berlin-New York (1970)Google Scholar
  160. 160.
    Zolotarëv, E.I.: A question on least quantities (1868). In: Zolotarëv, E.I.: Selected Papers, Vol. 2, pp. 130–166, USSR Academy of Sciences, Leningrad (1932). (in Russian)Google Scholar
  161. 161.
    Zolotarëv, E.I.: The theory of integral complex numbers with applications to integration (D.Sc. Thesis, 1874). In: Zolotarëv, E.I.: Selected Papers, Vol. 1, pp.161–360, USSR Academy of Sciences, Leningrad (1932). (in Russian)Google Scholar
  162. 162.
    Zolotarëv, E.I.: Applications of elliptic functions to questions of functions deviating least and greatest from zero (1877). In: Zolotarëv, E.I.: Selected Papers, Vol. 2, pp. 1–59, USSR Academy of Sciences, Leningrad (1932). (in Russian)Google Scholar
  163. 163.
    Zvonkine, D.A., Lando, S.K.: On multiplicities of the Lyashko-Looijenga mapping on discriminant strata. Funktsonal. Anal. i Prilozhen. 33:3, 21–34 (1999). English transl. in Funct. Anal. Appl. 33:3, 178–188 (1999)Google Scholar

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© Springer-Verlag GmbH Berlin Heidelberg 2012

Authors and Affiliations

  • Andrei Bogatyrev
    • 1
  1. 1.Institute of Numerical Mathematics of the Russian Academy of SciencesMoscowRussia

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