Constructive Approximation

, Volume 11, Issue 1, pp 85–106

Incomplete rational approximation in the complex plane

  • P. B. Borwein
  • Weiyu Chen
Article

Abstract

We consider rational approximations of the form
$$\left\{ {(1 + z)^{\alpha n + 1} \frac{{p_{cn} (z)}}{{q_n (z)}}} \right\}$$
in certain natural regions in the complex plane wherepcn andqn are polynomials of degreecn andn, respectively. In particular we construct natural maximal regions (as a function of α andc) where the collection of such rational functions is dense in the analytical functions. So from this point of view we have rather complete analog theorems to the results concerning incomplete polynomials on an interval.

The analysis depends on an examination of the zeros and poles of the Padé approximants to (1+z)αn+1. This is effected by an asymptotic analysis of certain integrals. In this sense it mirrors the well-known results of Saff and Varga on the zeros and poles of the Padé approximant to exp. Results that, in large measure, we recover as a limiting case.

In order to make the asymptotic analysis as painless as possible we prove a fairly general result on the behavior, inn, of integrals of the form
$$\int_0^1 {[t(1 - t)f_z (t)]^n {\text{ }}dt,}$$
wherefz(t) is analytic inz and a polynomial int. From this we can and do analyze automatically (by computer) the limit curves and regions that we need.

AMS classification

Primary 41A20, 41A21 Secondary 41A60, 30C15 

Key words and phrases

Padé approximation Incomplete rationals Incomplete polynomials Steepest descent Zeros Poles 

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References

  1. 1.
    R. B. Ash (1971): Complex Variables. New York: Academic Press.Google Scholar
  2. 2.
    P. Borwein, W. Chen, K. Dilcher (to appear):Zeros of iterated integrals of polynomials. Canad. J. Math.Google Scholar
  3. 3.
    G. G. Lorentz (1980):Problems for incomplete polynomials. In: Approximation Theory III (E. W. Cheney, ed.). New York: Academic Press, pp. 41–73.Google Scholar
  4. 4.
    M. Marden (1966): Geometry of Polynomials. Mathematical Surveys, no. 3. Providence, RI: American Mathematical Society.Google Scholar
  5. 5.
    F. W. J. Olver (1974): Asymptotics and Special Functions. New York: Academic Press.Google Scholar
  6. 6.
    G. Pólya, G. Szegö (1976): Problems and Theorems in Analysis I. Berlin: Springer-Verlag.Google Scholar
  7. 7.
    E. B. Saff (1983):Incomplete and orthogonal polynomials. In: Approximation Theory IV (C. K. Chui, L. L. Schumaker, J. D. Ward, eds.), New York: Academic Press, pp. 219–256.Google Scholar
  8. 8.
    E. B. Saff, R. S. Varga (1975):On the zeros and poles of Padé approximants to e z. Numer. Math.,25:1–14.Google Scholar
  9. 9.
    E. B. Saff, R. S. Varga (1977):On the zeros and poles of Padé approximations to e z,II. In: Padé and Rational Approximants: Theory and Applications (E. B. Saff, R. S. Varga, eds.). New York: Academic Press, pp. 195–213.Google Scholar
  10. 10.
    E. B. Saff, R. S. Varga (1978):On the zeros and poles of Padé approximants to e z,III. Numer. Math.,30:241–266.Google Scholar
  11. 11.
    G. Szegö (1924):Über eine Eigenschaft der Exponentialreihe. Sitzungsber. Berl. Math. Ges.,23:50–64.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1995

Authors and Affiliations

  • P. B. Borwein
    • 1
  • Weiyu Chen
    • 2
  1. 1.Department of Mathematics and StatisticsSimon Fraser UniversityBurnabyCanada
  2. 2.Department of Mathematical ScienceUniversity of AlbertaEdmontonCanada

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