Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T22:18:30.084Z Has data issue: false hasContentIssue false
  • English
  • Français

Automorphic Orthogonal and Extremal Polynomials

Published online by Cambridge University Press:  20 November 2018

A. L. Lukashov  and
F. Peherstorfer
A. L. Lukashov
Affiliation:
Institut für Mathematik, Johannes Kepler Universität Linz, A-4040 Linz, Austria e-mail: [email protected]
F. Peherstorfer
Affiliation:
Institut für Mathematik, Johannes Kepler Universität Linz, A-4040 Linz, Austria e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szegö polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szegö polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients etc. are derived.

Keywords

42C0530F3531A1541A2141A50
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 3 , 01 June 2003 , pp. 576 - 608
Copyright
Copyright © Canadian Mathematical Society 2003

References

[1] Abel, N. H. Über die Integration der differential-Formeln , wenn R und ρ ganze Funktionen sind. J. Reine Angew. Math. 1(1826), 186–221.Google Scholar
[2] Akhieser, N. I., Über einige Funktionen, die in gegebenen Intervallen am wenigsten von Null abweichen. Bull. Soc. Phys. -Math. Kazan (3) (2)3(1928), 1–69.Google Scholar
[3] Akhieser, N. I., Vorlesungen über Approximationstheorie. Akademie Verlag, Berlin, 1953.Google Scholar
[4] Akhieser, N. I. Orthogonal polynomials on several intervals. Soviet Math. Dokl. 1(1960), 989–992.Google Scholar
[5] Akhieser, N. I. and Yu. Ya. Tomčuk On the theory of orthogonal polynomials over several intervals. Soviet Math. Dokl. 2(1961), 687–690.Google Scholar
[6] Aptekarev, A. I. Asymptotic properties of polynomials orthogonal on a system of contours, and periodic motions of Toda lattices. Mat. Sb. 53(1986), 233–260.Google Scholar
[7] Baker, H., Abel's theorem and the allied theory. 2nd ed. Cambridge Univ. Press, Cambridge, UK, 1995.Google Scholar
[8] Bateman, H. and A. Erdélyi, Higher transcendental functions. Vol. 2. McGraw-hill, New York, NY, 1955.Google Scholar
[9] Belokolos, E. D., Bobenko, A. I., Enol'skii, V. Z., Its, A. R. and Matveev, V. B., Algebro-geometric approach to nonlinear integrable equations. Springer, Berlin, 1994.Google Scholar
[10] Bernstein, S. N. Sur les polynomes orthogonaux relatifs à un segment fini, I. J. Math. Pure Appl. 9(1930), 127–177.Google Scholar
[11] Bogatyrev, A. B. On the efficient computation of Chebyshev polynomials for several intervals. Sb. Math. 190(1999), 1571–1605.Google Scholar
[12] Borwein, P. and T. Erdélyi, Polynomials and Polynomial Inequalities. Springer, Berlin, 1995.Google Scholar
[13] Burnside, W. On a class of automorphic functions. Proc. London Math. Soc. 23(1892), 49–88.Google Scholar
[14] Burnside, W. Further note on automorphic functions. Proc. London Math. Soc. 23(1892), 281–295.Google Scholar
[15] Draux, A., Polyn.omes orthogonaux formels.applications. Springer, Berlin, 1983.Google Scholar
[16] Falliero, T. and Sebbar, A. Capacité d'une union de trois intervalles et fonctions thêta de genre 2. J. Math. Pure Appl. 80(2001), 409–443.Google Scholar
[17] Ford, L. R., Automorphic functions. McGraw-hill, New York, 1929.Google Scholar
[18] Geronimo, J. S. and Johnson, R. An inverse problem associated with polynomials orthogonal on the unit circle. Comm. Math. Phys. 193(1998), 125–150.Google Scholar
[19] Geronimus, Ya. L., On some finite difference equations and corresponding systems of orthogonal polynomials. Zap. Mat. Otdel. Fiz.-Mat. Fak. i Khar'kov Mat. Obshch., Ser. 4. 25(1957), 87–100 [Russian].Google Scholar
[20] Gončar, A. A. On the convergence of generalized Padé approximants of meromorphic functions. Mat. Sb. 97(1975), 607–629.Google Scholar
[21] Halphen, G. H., Traité des fonctions elliptiques et leurs applications. tom. II. Gauthier-Villars, Paris, 1888.Google Scholar
[22] Hlawka, E., Schoissengeier, J. and Tascher, R., Geometric and analytic number theory. Springer, Berlin, 1991.Google Scholar
[23] Jacobi, C. G. J. Note sur une nouvelle application de l'analyse des fonctions elliptiques à l'algèbre. J. Reine Angew. Math. 7(1830), 41–43.Google Scholar
[24] Krein, M. G., Levin, B. Ya., and Nudel'man, A. A., On special representation of polynomials that are positive on a system of closed intervals, and some applications. In: Functional Analysis, Optimization, and Mathematical Economics: A Collection of Papers Dedicated to the Memory of Leonid Vital'evich Kantorovich, (ed., L. J. Leifman), Oxford Univ. Press, New York, 1990, 56–114.Google Scholar
[25] Lebedev, V. I., Extremal polynomials with restrictions and optimal algorithms. In: Advanced Mathematics: Computations and Applications, (eds., A. S. Alekseev and N. S. Bakhvalov), NCC Publ., Novosibirsk, 1995, 491–502.Google Scholar
[26] Lebedev, V. I. On the solution of inverse problems and trigonometric forms for the Geronimus polynomials. Application to the theory of iterative methods. Russian J. Numer. Anal. Math. Modelling 15(2000), 73–93.Google Scholar
[27] Lubinsky, D. S., Zeros of orthogonal and biorthogonal polynomials: some old, some new. In: Nonlinear numerical methods and rational approximation, II, (ed., A. Cuyt), Kluwer Acad. Publ., Dordrecht, 1994, 3–15.Google Scholar
[28] Lubinsky, D. S., Will Ramanujan kill Baker-Gammel-Wills? (A selective survey of Padé approximation). In: New developments in approximation theory, (eds., M.W. Müller et al.), Birkhäuser, Basel, 1999, 159–174.Google Scholar
[29] Lubinsky, D. S. Asymptotics of orthogonal polynomials: some old, some new, some identities. Acta. Appl. Math. 61(2000), 207–256.Google Scholar
[30] Lukashov, A. L., The algebraic fractions of Chebyshev and Markov on several segments. Anal. Math. 24(1998), 111–130 [Russian].Google Scholar
[31] Lukashov, A. L. On Chebyshev-Markov rational fractions over several intervals. J. Approx. Theory 95(1998), 333–352.Google Scholar
[32] Magnus, A. Recurrence coefficients for orthogonal polynomials on connected and nonconnected sets. In: Padé Approximation and its Application, Lecture Notes in Math. 765(1979), 150–171.Google Scholar
[33] Nuttall, J. and Singh, S. R. Orthogonal polynomials and Padé approximations associated with a system of arcs. J. Approx. Theory 21(1977), 1–42.Google Scholar
[34] Peherstorfer, F. On Bernstein-Szegö orthogonal polynomials on several intervals. SIAM J. Math. Anal. 21(1990), 461–482.Google Scholar
[35] Peherstorfer, F. On Bernstein-Szegö orthogonal polynomials on several intervals,II: Orthogonal polynomials with periodic recurrence coefficients. J. Approx. Theory 64(1991), 123–161.Google Scholar
[36] Peherstorfer, F. Orthogonal and extremal polynomials on several intervals. J. Comput. Appl. Math. 48(1993), 187–205.Google Scholar
[37] Peherstorfer, F., Positive and orthogonal polynomials on several intervals. Rend. Circ. Mat. Palermo (2) (33)(1993), 399–414.Google Scholar
[38] Peherstorfer, F. Elliptic orthogonal and extremal polynomials. Proc. London Math. Soc. (3) 70(1995), 605–624.Google Scholar
[39] Peherstorfer, F., Minimal polynomials on several intervals with respect to the maximum-norm.a survey. In: Complex methods in approximation theory, (eds., A. M. Finkelshtein et al.), Univ. Alméria, Alméria, 1997, 137–159.Google Scholar
[40] Peherstorfer, F. and Schiefermayr, K., Description of extremal polynomials on several intervals and their computation, I, II. Acta Math. Hungar. 83(1999), 27-58, 59–83.Google Scholar
[41] Peherstorfer, F. and Steinbauer, R. On polynomials orthogonal on several intervals. Ann. of Numer. Math. 2(1995), 353–370.Google Scholar
[42] Ptashitskiĭ, I. L., On integration in finite form of irrational differentials. Saint-Petersburg, Thesis, 1881. [Russian]Google Scholar
[43] Rusak, V. N., Rational functions as an apparatus of approximation. Byelorussian Univ. Press, Minsk, 1979. [Russian]Google Scholar
[44] Shen, J., Strang, G. and Wathen, A. J. The potential theory of several intervals and its applications. Appl. Math. Optim. 44(2001), 67–85.Google Scholar
[45] Shohat, J. On the continued fractions associated with, and corresponding to, the integral . Amer. J. Math. 55(1933), 218–230.Google Scholar
[46] Sodin, M. L. and Yuditskiĭ, P. M. Functions deviating least from zero on closed subsets of the real axis. St. PetersburgMath. J. 4(1993), 201–249.Google Scholar
[47] Sodin, M. L. and Yuditskiĭ, P. M. Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. Geom. Anal. 7(1997), 387–435.Google Scholar
[48] Stahl, H. The convergence of Padé approximants to functions with branch points. J. Approx. Theory 91(1997), 139–204.Google Scholar
[49] Suetin, S. P. Uniform convergence of Padé diagonal approximants for hyperelliptic functions. Sb. Math. 191(2000), 1339–1373.Google Scholar
[50] Szegö, G., Orthogonal polynomials. 4th ed. Amer.Math. Soc., Providence, RI, 1975.Google Scholar
[51] Ya, Yu.. Tomčuk, Orthogonal polynomials on a system of intervals of the real axis. Zap. Mekh.-Mat. Fak. i Khar'kov Mat. Obshch., Ser. 4 29(1964), 93–128 [Russian].Google Scholar
[52] Tsuji, M., Potential theory in modern function theory. 2nd ed. Chelsea Publ., New York, NY, 1975.Google Scholar
[53] Vasiliev, R. K., Chebyshev polynomials and approximation theory on compact subsets of the real axis. Saratov Univ. Publ., Saratov, 1998.Google Scholar
[54] Widom, H. Extremal polynomials associated with a system of curves in the complex plane. Adv. in Math. 3(1969), 127–232.Google Scholar