Advertisement

Journal d’Analyse Mathématique

, Volume 70, Issue 1, pp 225–266 | Cite as

A criterion for uniqueness of a critical point inH 2 rational approximation

  • L. Baratchart
  • E. B. Saff
  • F. Wielonsky
Article

Abstract

This paper presents a criterion for uniqueness of a critical point inH 2,R rational approximation of type (m, n), withmn-1. This criterion is differential-topological in nature, and turns out to be connected with corona equations and classical interpolation theory. We illustrate its use with three examples, namely best approximation of fixed type on small circles, a de Montessus de Ballore type theorem, and diagonal, approximation to the exponential function of large degree.

Keywords

Rational Function Rational Approximation Morse Index Critical Pair Real Polynomial 
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.

Notations

T,U,V

unit circle, open unit disk, complement in\(\bar C\) of the closed unit disk

Tr,Ur,Vr

circle of radiusr, open disk of radiusr, complement in\(\bar C\) of the closed disk of radiusr (with centers at the origin)

Pn

space of real polynomials of degree at mostn; regarding the coefficients as coordinates, we endowP n with the Euclidean topology ofR n+1

Mn

monic real polynomials of degreen

\(\bar {\mathcal{M}}_n \)

real polynomials of degree at mostn with constant coefficient equal to 1

\(\mathcal{M}_n^r \)

monic real polynomials of degreen having all their roots inU r

\(\bar {\mathcal{M}}_n^r \)

real polynomials of degree at mostn with constant coefficient equal to 1 having all their roots inV r

Δn

real monic polynomials of degreen having all their roots in\(\bar U\) alternatively closure ofM n 1 with respect to the Euclidean topology ofP n

\(\tilde \Delta _n \)

real polynomials of degree at mostn with constant coefficient equal to 1 having all their roots in\(\bar V\); alternatively, closure of\(\widetilde{\mathcal{M}}_n^1 \) with respect to the Euclidean topology ofP n

‖·‖∞,‖·‖2

norms inL (T) and inL 2(T), respectively

<·,·>

scalar product inL 2(T)

L2,R(T)

real subspace ofL 2(T) consisting of functions with real Fourier coefficients

H2,R(U)

real Hardy space of exponent 2 of the unit disk consisting of functions inL 2,R(T) whose Fourier coefficients with negative index vanish

H2,R0(V)

real Hardy space of exponent 2 of the complement of the closed unit disk restricted to those functions vanishing at infinity; alternatively, orthogonal complement ofH 2,R (U) inL 2,R (T)

P+,P

orthogonal projectionsL 2,R (T)→H 2,R (U) andL 2,R (T)→H 2,R 0 (V), respectively

H∞,R(U)

real subspace ofH 2,R (U) consisting of essentially bounded functions

\(\mathcal{R}_{m,n}^0 \left( V \right)\)

subset ofH 2,R 0 (V) consisting of rational functionsp/z m-n+1 q withpP m andqM n 1

\(\widetilde{\mathcal{R}}_{m,n}^0 \left( U \right)\)

subset ofH 2,R (U) consisting of rational functions\(\tilde p/\tilde q\) with\(\tilde p \in \mathcal{P}_m \) and\(\tilde q \in \widetilde{\mathcal{M}}_n^1 \)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. A. Baker, Jr.,Essentials of Padé Approximants, Academic Press, New York, 1975.MATHGoogle Scholar
  2. [2]
    L. Baratchart,Sur l'approximation, rationnelle l 2 pour les systèmes dynamiques linéaires, Thèse de doctorat d'état, Université de Nice, 1987.Google Scholar
  3. [3]
    L. Baratchart and M. Olivi,Index of critical points in l 2-approximation, Systems & Control Letters10 (1988), 167–174.MATHCrossRefGoogle Scholar
  4. [4]
    L. Baratchart, M. Olivi and F. Wielonsky,On a rational approximation problem in the real Hardy space H 2. Theoretical Computer Science94 (1992), 175–197.MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    L. Baratchart, E. B. Saff and F. Wielonsky,Rational interpolation of the exponential function, Canad. J. Math.47 (1995), 1121–1147.MATHMathSciNetGoogle Scholar
  6. [6]
    L. Baratchart and F. Wielonsky,Rational approximation in the real Hardy space H 2 and Stieltjes integrals: a uniqueness theorem, Constr. Approx.9 (1993), 1–21.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    D. Braess,Nonlinear Approximation Theory, Springer Series in Computational Mathematics, Vol. 7, Springer-Verlag, Berlin, 1986.MATHGoogle Scholar
  8. [8]
    P. E. Caines,Linear Stochastic Systems, Probability and Mathematical Statistics, Wiley, New York, 1988.Google Scholar
  9. [9]
    J. Della Dora,Contribution à l'approximation de fonctions de la variable complexe au sens de Hermite-Padé et de Hardy, Thèse d'état, Univ. Scient. et Medicale de Grenoble, 1980.Google Scholar
  10. [10]
    J. C. Doyle, B. A. Francis and A. R. Tannenbaum,Feedback Control Theory, Macmillan, New York 1992.Google Scholar
  11. [11]
    M. Duc-Jacquet,Approximation des fonctionelles linéaires sur les espaces Hilbertiens à noyaux reproduisants, Thèse d'état, Univ. Scient. et Medicale de Grenoble, 1973.Google Scholar
  12. [12]
    V. D. Erohin,On the best approximation of analytic functions by rational functions, Doklady Akad. Nauk SSSR128 (1959), 29–32 (in Russian).MATHMathSciNetGoogle Scholar
  13. [13]
    S. Lang,Complex Analysis, Addison-Wesley, Reading, MA, 1977.MATHGoogle Scholar
  14. [14]
    A. L. Levin,The distribution of poles of rational functions of best approximation and related questions, Math. USSR Sbornik9, No. 2 (1969), 267–274.MATHCrossRefGoogle Scholar
  15. [15]
    L. Ljung,System Identification: Theory for the User, Prentice-Hall, New Jersey, 1987.MATHGoogle Scholar
  16. [16]
    D. J. Newman,Approximation with Rational Functions, Regional Conf. Series41, AMS, Providence, Rhode Island, 1979.MATHGoogle Scholar
  17. [17]
    O. Perron,Die Lehre von den Kettenbrüchen, 3rd ed., Vol. 2, B. G. Teubner, Stuttgart, 1957.MATHGoogle Scholar
  18. [18]
    P. P. Petrushev and V. A. Popov,Rational Approximation of Real Functions, inEncyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1987.Google Scholar
  19. [19]
    G. Pólya and G. Szegö,Problems and Theorems in Analysis, I, Springer-Verlag, Berlin, 1972.Google Scholar
  20. [20]
    P. A. Regalia,Adaptative IIR Filtering in Signal Processing and Control, Dekker, New York, 1995.Google Scholar
  21. [21]
    E. B. Saff,An extension of Montessus de Ballore's theorem on the convergence of interpolating rational functions, J. Approx. Theory6 (1972), 63–67.MATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    E. B. Saff and R. S. Varga,On the zeros and poles of Padé approximants to e z.II, inPadé and Rational Approximations: Theory and Applications (E. B. Saff and R. S. Varge, eds.), Academic Press, New York, 1977, pp. 195–213.Google Scholar
  23. [23]
    T. Söderström,On the uniqueness of maximum likelihood identification, Automatica11 (1975), 193–197.MATHCrossRefGoogle Scholar
  24. [24]
    L. N. Trefethen,The asymptotic accuracy of rational approximations to e z on a disk, J. Approx. Theory40 (1984), 380–383.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    M. Tsuji,Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.MATHGoogle Scholar
  26. [26]
    J. L. Walsh,Interpolation and Approximation by Rational Functions in the Complex Domain, AMS Colloq. Publ. XX, 1969.Google Scholar

Copyright information

© Hebrew University of Jerusalem 1996

Authors and Affiliations

  • L. Baratchart
    • 1
  • E. B. Saff
    • 2
  • F. Wielonsky
    • 1
  1. 1.INRIASophia Antipolis CedexFrance
  2. 2.ICM, Department of MathematicsUniversity of South FloridaTampaUSA

Personalised recommendations