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Applications of Transformation Theory: A Legacy from Zolotarev (1847–1878)

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Approximation Theory and Spline Functions

Part of the book series: NATO ASI Series ((ASIC,volume 136))

Abstract

What we are concerned with is, roughly, the generalization to the elliptic case of the familiar multiple angle formulas of elementary trigonometry such as

$$\begin{array}{l} \cos \,2\theta = 2{\cos ^2}\,\theta \, - \,1;\,\tan \,2\theta = \frac{{2\,\tan \theta }}{{1 - {{\tan }^2}\theta }};\\ \sin \,2\theta = 2\,\sin \,\theta \cos \, \in = 2\sin \,\theta \sqrt {\left( {1 - {{\sin }^2}\theta } \right)} \end{array}$$

(which are respectively polynomial, rational, algebraic). More generally we have

$$\cos \,{\rm{n}}\theta = {2^{n - 1}}\left[ {{{\cos }^n}\theta - \frac{1}{4}\,{{\cos }^{n - 2}}\,\theta \, + \, \ldots } \right]$$

which we can also express as a Chebyshev polynomial:

$${{\rm{T}}_{\rm{n}}}\left( {\rm{x}} \right) = {\rm{cos}}\left( {{\rm{n}}\,{\rm{arccos}}\,{\rm{x}}} \right) = {{\rm{2}}^{{\rm{n}} - {\rm{1}}}}\left[ {{{\rm{x}}^{\rm{n}}} - \frac{{\rm{1}}}{{\rm{4}}}{\rm{n}}{{\rm{x}}^{{\rm{n}} - {\rm{2}}}}\,{\rm{ + }}\,{\rm{ \ldots }}} \right] = {{\rm{2}}^{{\rm{n}} - {\rm{1}}}}{\rm{ }}{{\rm{\tilde T}}_{\rm{n}}}\left( {\rm{x}} \right)$$

“And out of olde bokes, in good feyth, Cometh all this newe science that men lere”

CHAUCER, THE PARLIAMENT OF FOWLS

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References

  1. Achiezer, N. I., “Vorlesungen liber Approximations-theorie”, Akademic Verlag, Berlin, 1953, translated as, “On the Theory of Approximation”, Ungar, New York, 1956. (Second German edition, Berlin, 1967.) “Elements of the Theory of Elliptic Functions”, 2nd edition (in Russian), 1970. For complete bibliographies of Achiezer see the articles on him in Uspehi Mat. Nauk ≡ Russian Math. Surveys, 16, 6, (102),1961; p. 14; 26,6, (162), 1971, pp. 257–261; 36; 4, (220), 1981, pp. 183–4.

    Google Scholar 

  2. Amer, R. A., Schwarz, H. R., “Contributions to the approximation problem of electrical filters”, Mitt. Inst. ang. Math., ETH, Zürich, #9, 1964.

    Google Scholar 

  3. Bateman Project, “Higher Transcendental Functions, II”, McGraw Hill, New York, 1953.

    Google Scholar 

  4. Bernstein, S., “Lecons sur les propriéteé exctrémales et la meilleure approximation des fonctions analytiques d’une variable réelle”, Gauthier-Villars, Paris, 1926. Chel., 1970.

    Google Scholar 

  5. Blum, E. K., “Numerical analysis and computation, theory and practice”, Addison-Wesley, 1972.

    MATH  Google Scholar 

  6. Borwein, J. M., and Borwein, P. B., “Elliptic integrals and approximations to π”, to appear; Cubic and Higher Order Algorithms for π, Canada Math. Bull. to appear; Explicit algebraic nth order approximations to π, this volume; The arithmetic geometric mean and fast computation of elementary functions, SIAM Rev., to appear.

    Google Scholar 

  7. Brent, R. P., “Fast multiple precision evaluation of elementary functions”, J. ACM 23, 1976, pp. 212–251.

    Article  MathSciNet  Google Scholar 

  8. Carlson, B. C., and Todd, J., “The degenerating behavior of elliptic functions”, SIAM J. Numerical Anal. 20, 1983, pp. 1120–1129; “Zolotarev’s First Problem - the best approximation by polynomials of degree ≤ n − 2 to xn − n E xn−1, in [0, 1]σ], Aq. Math., 1984, 26 (1983, pp. 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  9. Cauer, W., “Synthesis of linear communications networks”, McGraw Hill, New York, 1958, German edition 1934 “Bemerkung über eine Extremalaufgabe von E. Zolotareff” ZAMM 20, 1940, 358.

    Google Scholar 

  10. Cayley, A., “Elliptic functions”, Deighton Bell, Cambridge 1895.

    Google Scholar 

  11. Cellitti, C., “Nuova representazione della sosti tuzione lineare binaria primitua, Atti Reale Acc. deiLincei. Ser. V, 23, 1914, pp. 208–213.

    Google Scholar 

  12. Chebyshev, P. L., “Oeuvres, I, II”, St. Petersburg, 1899, Chelsea, New York.

    Google Scholar 

  13. “Sur les fractions algebriques…”, Bull. Soc. Math. France, 12, 1884, pp. 167–168 (≡II, 725).

    Google Scholar 

  14. Cody, W. J., “Double-precision square root for the CDC-300”, Comm. ACM 7, 1964, pp. 715–718.

    Article  MATH  Google Scholar 

  15. Copson, E. T., “Functions of a complex variable”, Oxford, 1935.

    Google Scholar 

  16. Curtis, A. R., “Tables of Jacobian elliptic functions whose arguments are rational fractions of the quarter period”, NPL Math. Tables, v. 7, H. M. Stationery Office, London, 1964.

    MATH  Google Scholar 

  17. De Boor, C., and Rice, J. R., “Chebyshev approximation by a π[(x − xi)/(x − si)] and application to ADI iteration”, J. SIAM, 11, 1963, pp. 159–169.

    MATH  Google Scholar 

  18. de la Vallée Poussin, C. J., “Lecons sur l’approximation des fonctions d’ une variable réelle”, Gauthier-Villars, Paris, 1919, Chelsea, 1970.

    Google Scholar 

  19. Dixon, A. C., “The elementary properties of elliptic functions”, Macmillan, London, 1894.

    MATH  Google Scholar 

  20. Dumas, S., “Sur Ie developpement des functions elliptiques en fractions continues”, Thesis, Zürich, 1908.

    Google Scholar 

  21. Darlington, S. D., “Synthesis of resistance 4-poles which produce prescribed insertion loss characteristics including special application to filter design”, J. Math. Phys., 18, 1939, pp. 257–353.

    MathSciNet  Google Scholar 

  22. Enneper, A., rev. F. Müller, “Elliptische Functionen, Theorie und Geschichte”, Halle, 1890.

    MATH  Google Scholar 

  23. Erdös, P., and Szegö, G., “On a problem of I. Schur”, Ann. of Math. 243, 1942, pp. 451–470.

    Article  MathSciNet  MATH  Google Scholar 

  24. Farhutcinova, R. F., “Approximate analytic formulas for coefficients of Zolotarev polynomials and the process of economization via Zolotarev polynomials”, p. 126–135 in Uniform approximations and the moment problem, ed. B. Ja. Zinger, Dal nevostocn. Naucn. Centro Akad. Nauk, SSSR, Vladivostok., 1977.

    Google Scholar 

  25. Fricke, R., “Lehrbuch der Algebra”, 1924.

    MATH  Google Scholar 

  26. Gaier, D., Todd, J., “On th e rate of convergence of optimal ADI processes”, Numer. Math. 9, 1967, pp. 452–459.

    Article  MathSciNet  MATH  Google Scholar 

  27. Galeev, E. H., “Zolotarev Problem in the Metric of L1 (−1,1)”, Mat. Zametki 17, 1975, pp. 13–20 = Math. Notes, 1975, pp. 9–13.

    MathSciNet  MATH  Google Scholar 

  28. Gastinel, N., “Sur le meilleur choix des paramètres de sur-relaxation”, Chiffres 5, 1962, pp. 109–126.

    MathSciNet  Google Scholar 

  29. Gautschi, W., “Computational methods in special functions”, pp. 1–98 in Theory and Application of Special Functions, Academic Press, 1975.

    Google Scholar 

  30. Glowatski, E., “Sechstellige Tafel der Cauer-Parameter”, Abh. Bayer. Akad. Wiss., Math. Nat. Klasse, 67, 1955.

    Google Scholar 

  31. Goncar, A. A., “The problems of E.I. Zolotarev which are connected with rational functions”, Math. Sb. (N.S.), 78, 120, 1969, pp. 640–654 translated in USSR - Sb. 7, 1969, pp. 623–625.

    MathSciNet  Google Scholar 

  32. Greenhill, G., “The Applications of Elliptic Functions”, Macmillan, London, 1892.

    MATH  Google Scholar 

  33. Hornecker, G., “Évaluation approcheé de la meilleure approximation polynomiale d’ordre n de f(x) sur un segment fini [a,b]”, Chiffres 1, 1958, pp. 157–169.

    MathSciNet  Google Scholar 

  34. Houzel, C., “Fonctions elliptiques et intégrales abéliennes”, in Abrégé d’histoire des mathématiques 1700–1900, J. Dieudonné, directeur de la publication, 2 vols., Hermann, Paris, 1978.

    Google Scholar 

  35. King, L. V., “On the direct numerical calculation of elliptic functions and integrals”, Cambridge, 1924.

    Google Scholar 

  36. Kuznetsov, P. I., “D. F. Egorov (on the centenary of his birth)”, Usp. mat. nauk. = Russian Math. Surveys 26, 1971, #5, pp. 124–164.

    Google Scholar 

  37. Lang, S., “Elliptic Functions”, Addison-Wesley, Reading, Mass., 1973.

    MATH  Google Scholar 

  38. Lebedev, V. I., “On a Zolotarev problem in the method of alternating directions”, Z. Vycisl. Mat. i Mat. Fiz., 17, 1977, pp. 349–366, translated in USSR Compo Math. and Math. Phys., 17, 1977, pp. 58–76, 1978.

    MathSciNet  MATH  Google Scholar 

  39. Lebesgue, H., “Review of de la Vallee Poussin [1919]”, Bull. Sc. Math. 244, 1920, pp. 137–153 ≡ Oeuvres Scientifiques 5, 1973, pp. 322–338.

    Google Scholar 

  40. Magnus, W., Oberhettinger, F., and Soni, R. P., “Formulas and theorems for the special functions of mathematical physics”, 3, Springer, Heidelberg, 1966.

    MATH  Google Scholar 

  41. Markushevich, A. I., “The remarkable sine functions”, Elsevier, New York, 1966.

    MATH  Google Scholar 

  42. Meiman, N. N., “Solution of the fundamental problems of the theory of polynomials and entire functions which deviate least from zero”, (in Russian), Trudy Moskov. Math. Obsc. 9, 1960, pp. 507–535, MR24A#2031.

    MathSciNet  Google Scholar 

  43. Meiman, N. N., “On the theory of polynomials deviating least from zero”, 1960, translated in Soviet Math. Dokl. 1, pp. 41–44.

    MathSciNet  MATH  Google Scholar 

  44. Meiman, N. N., “Polynomials deviating least from zero with an arbitrary number of given coefficients”, 1960, translated in Soviet. Math. Dokl. 1, pp. 72–75.

    MathSciNet  MATH  Google Scholar 

  45. Meiman, N. N., “The zeros of a class of multiple valued-functions”, translated in AMS Trans. 2, 19, 1962, pp. 167–171.

    MathSciNet  Google Scholar 

  46. Meiman, N. N., “The theory of functions of class HB and Bx”, translated in AMS Trans. (2), 19, 1962, pp. 173–178.

    MathSciNet  Google Scholar 

  47. Meinardus, G., “Approximation von Funktionen und ihre numerische Behand1ung”, Springer, 1964, translated by L. L. Schumaker as “Approximation of Functions, Theory and Numerical Methods”, Springer, 1967.

    Google Scholar 

  48. Meinardus, G., and Taylor, G. D., “Optimal partitioning of Newton’s method for calculating roots”, Math. Comp. 35, 1980, pp. 1231–1250.

    Article  MathSciNet  Google Scholar 

  49. Meinguet, J., and Belevitch, V., “On the realizability of ladder filters”, IRE Trans. Prof. Groups on Circuit Theory, CT 5, 1958, pp. 253–255.

    Google Scholar 

  50. Melzak, Z. A., “Mathematical ideas, modeling and applications”, Vol. II of Companion to concrete mathematics, Wiley, New York, 1976.

    Google Scholar 

  51. Milne-Thomson, L. M., “Jacobian elliptic function tables”, Dover, New York, 1950.

    MATH  Google Scholar 

  52. Moursund, D. G., “Optimal starting values for Newton-Raphson calculations of √x”, Comm. ACM 10, 1967, pp. 430–432.

    Article  MathSciNet  Google Scholar 

  53. Moursund, D. G., and Taylor, G. D., “Optimal starting values for the Newton-Raphson calculation of inverses of certain functions”, SIAM J. Numer. Anal. 5, 1968, pp. 138–150.

    Article  MathSciNet  MATH  Google Scholar 

  54. NBS Handbook of Mathematical Functions, ed. M. Abramowitz - I. A. Stegun, U. S. Government Printing Office, Washington, D.C., 1964.

    Google Scholar 

  55. Neville, E. H., “Jacobian elliptic functions”, Oxford, 1946.

    Google Scholar 

  56. Newman, D. J., “Approximation with rational functions”, CBMS, Regional conference series in mathematics #41, 1978, American Math. Soc., Providence, R.I.

    Google Scholar 

  57. Newman, D. J., “Rational approximation versus fast computer methods”, in Lectures on approximation and value distribution, Univ. de Montréal, 1982.

    Google Scholar 

  58. Ninomiya, I., “Generalized rational Chebyshev approximation”, Math. Compo 24, 1970, pp. 159–169. “Best rational starting approximations and improved Newton iteration for the square roots”, Math. Comp. 24, 1970, pp. 391–404.

    Article  MathSciNet  MATH  Google Scholar 

  59. Oberhettinger, F., and Magnus, W., “Anwendung der elliptischen Funktionen in Physik und Technik”, Springer-Verlag, Heidelberg, 1949.

    MATH  Google Scholar 

  60. Paszkowski, S., “The theory of uniform approximation”, I, Rozprawy Matem., 25, Warsaw, 1962.

    Google Scholar 

  61. Piloty, H., “Zolotareffsche rationale Funktionen”, ZAMM 34, 1954, pp. 175–189.

    Article  MathSciNet  MATH  Google Scholar 

  62. Ozigova, E. P., “Zolotarev, E. I., 1847–1878”, Nauka, Moscow, 1966 (in Russian).

    Google Scholar 

  63. Peaceman, D. W., and Rachford, H. H., Jr., “The numerical solution of parabolic and elliptic differential equations”, J. SIAM 3, 1955, pp. 28–41.

    MathSciNet  MATH  Google Scholar 

  64. Feherstorfer, F., “On the representation of extremal functions in the L1-norm”, J. Approx. Theory, 27, 1979, pp. 61–75.

    Article  MathSciNet  Google Scholar 

  65. Pokrovskii, V. L., “On a class of polynomials with extremal properties”, Mat. Sb. (N.S.) 48, 90, pp. 257–276, translated in AMS Trans. (2), 19, 1962, pp. 199–219.

    Google Scholar 

  66. Rauch, H. E., and Leibowitz, A., “Elliptic functions, theta-functions and Riemann surfaces”, Baltimore, 1973.

    MATH  Google Scholar 

  67. Rivlin, T. J., “Polynomials of best approximation to certain rational functions”, Numer. Math. 4, 1962, pp. 345–369.

    Article  MathSciNet  MATH  Google Scholar 

  68. Robert, Alain, “Elliptic curves”, Springer Lecture Notes in Mathematics, #326, 1973.

    MATH  Google Scholar 

  69. Reddy, A. R., “Rational approximation to xn+1 − σxn”, Quart. J. Math. (Oxford), 2, 28, 1977, pp. 123–127.

    Article  Google Scholar 

  70. Reddy, A. R., “A note on a result of Zolotarev and Bernstein”, Manuscripta Math. 10, 1977, pp. 95–97.

    Article  Google Scholar 

  71. Reddy, A. R., “On certain problems of Chebyshev, Zolotarev, Bernstein and Akhieser”, Invent. Math. 45, 1978, pp. 83–110.

    Article  MathSciNet  Google Scholar 

  72. Rice, J. R., “The approximation of functions”, 2 vols., Addison-Wesley, 1964, 1969.

    MATH  Google Scholar 

  73. Riemann, B., “Elliptische Functionen”, ed. H. Stahl, Teubner, Leipzig, 1899.

    Google Scholar 

  74. Rivlin, T. J., “An introduction to the approximation of functions”, 1969, 1981. Dover, N.Y.

    Google Scholar 

  75. Ryzakov, I. Ju., “An analog of a problem of E. I. Zolotarev”, DAN USSR 160, 1965, 552–559 ≡ Soviet Math. Dokl. 6, 1965, pp. 157–159.

    Google Scholar 

  76. Ryzakov, I. Ju., “The trigonometrical analogue of a problem of E. I. Zolotarev”, Izv. Vyss. Ucebn. Zaved. Mat. 8, 87, 1969, pp. 75–88.

    Google Scholar 

  77. Saff, E. B., and Varga, R. S., “Remarks on some conjectures of G. G. Lorentz”, J. Approx. Theory 30, 1980, pp. 29–36.

    Article  MathSciNet  MATH  Google Scholar 

  78. Salamin, E., “Computation of π using arithmetic-geometric mean”, Math. Compo 30, 1976, pp. 565–570.

    MathSciNet  MATH  Google Scholar 

  79. Stiefel, E. L., “Le problème d’approximation dans la theorie des filtres electriques”, pp. 81–87, Belgian Colloquium, 1961.

    Google Scholar 

  80. Talbot, A., “On a class of Tchebysheffian approximation problems solvable algebraically”, Proc. Cambridge Phil. Soc. 58, 1962, pp. 266–267.

    Article  MathSciNet  Google Scholar 

  81. Talbot, A., “The Tchebysheffian approximation of one rational function by another”, Proc. Cambridge Phil. Soc. 60, 1964, pp. 877–890.

    Article  MathSciNet  MATH  Google Scholar 

  82. Talbot, A., Inaugural Lecture, “Approximation Theory or a miss is better than a mile”, 18 November 1970, University of Lancaster Library, 1971.

    Google Scholar 

  83. Todd, J., “Introduction to the constructive theory of functions”, Birkhauser, Academic Press, 1963.

    MATH  Google Scholar 

  84. Todd, J., “Optimal ADI-parameters”, in Functional Analysis, Approximationstheorie, Numerische Mathematik, ISNM, 7, Birkhäuser, Basel, 1967.

    Google Scholar 

  85. Todd, J., “Inequalities of Chebyshev, Zolotarev, Cauer and W. B. Jordan”, pp. 321–328 in Inequalities, II., ed. O. Shisha, Academic Press, 1967.

    Google Scholar 

  86. Todd, J., “Optimal parameters in two programs”, Bull. Inst. Math. Appl. 6, 1970, pp. 31–35.

    Google Scholar 

  87. Todd, J., “The lemniscate constants”, Comm. ACM 18, 1975, pp. 16–19.

    Article  MathSciNet  Google Scholar 

  88. Todd, J., “The many values of mixed means”, I., pp. 5–22, in General Inequalities, ed. E. F. Beckenbach, ISNM 41, Birkhäuser, Basel, 1978.

    Google Scholar 

  89. Todd, J., “Some applications of elliptic functions and integrals”, Coll. Math. Soc. Janos Bolyai, vol. 22, Numerical Methods, ed. P. Rózsa, Keszthely (Hungary), 1977, pp. 591–618, North Holland, New York, and Amsterdam, 1980.

    MathSciNet  Google Scholar 

  90. Todd, J., “The best polynomial approximation to (1+x)−1 in [0,1]”, to appear.

    Google Scholar 

  91. Tricomi, F. G., and Kraft, M., “Elliptische Funktionen”, Leipzig, 1948.

    MATH  Google Scholar 

  92. Varga, R. S., “Matrix iterative analysis”, Prentice-Hall, Englewood Cliffs, 1962.

    Google Scholar 

  93. Voronowskaja, E. V., “The functional method and its applications”, Trans. Russian Math. Monographs, 28, 1970, American Math. Soc., Providence, R.I.

    Google Scholar 

  94. Wachspress, E. L., “Extended application of ADI iteration model problem theory”, J. SIAM 11, 1963, pp. 994–1016.

    MathSciNet  Google Scholar 

  95. Wachspress, E. L., “Optimum ADI iteration parameters for a model problem”, J. SIAM 10, 1962, pp. 339–350.

    MathSciNet  MATH  Google Scholar 

  96. Wachspress, E. L., “Iterative solution of elliptic systems”, Prentice-Hall, Englewood Cliffs, 1966.

    MATH  Google Scholar 

  97. Whittaker, E. T., and Watson, G. N., “Modern analysis”, Cambridge Univ. Press, 1927.

    MATH  Google Scholar 

  98. Zolotarev, E. I., “Complete collected works”, I, 1931, II, 1932, Leningrad.

    Google Scholar 

  99. Zolotarev, E. I., “Sur l’ application des functions elliptiques aux questions de maxima et minima”, Bull. Acad. Sc. St. Petersburg 3, 24, 1878, 305–310, Melanges, 5, pp. 419–426 (≡I, pp. 369–376).

    Google Scholar 

  100. Zolotarev, E. I., “Anwendungen den elliptischen Functionen auf Probleme über Functionen die von Null am wenigsten oder am meisten abweichen”(in Russian), Abh. St. Petersburg 30, 1877, S. (≡II, 1–59). See also remarks by N. G. Chebotarev and N. I. Achieser, II, pp. 357–361, Ob odnom voprose o naimehlshix velichnax, Diss. 1868, (≡II, 130–166).

    Google Scholar 

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  1. California Institute of Technology, USA

    John Todd

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  1. Department of Mathematics and Statistics, Memorial University, St. John’s, Newfoundland, Canada

    S. P. Singh , J. W. H. Burry  & B. Watson ,  & 

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Todd, J. (1984). Applications of Transformation Theory: A Legacy from Zolotarev (1847–1878). In: Singh, S.P., Burry, J.W.H., Watson, B. (eds) Approximation Theory and Spline Functions. NATO ASI Series, vol 136. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6466-2_11

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