Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Products of polynomials in uniform norms


Author: Igor E. Pritsker
Journal: Trans. Amer. Math. Soc. 353 (2001), 3971-3993
MSC (1991): Primary 30C10, 11C08, 30C15; Secondary 31A05, 31A15
DOI: https://doi.org/10.1090/S0002-9947-01-02856-2
Published electronically: May 21, 2001
MathSciNet review: 1837216
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment $[-1,1]$. Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set of positive logarithmic capacity in the complex plane. The above classical results are contained in our theorem as special cases.

It is shown that the asymptotically extremal sequences of polynomials, for which this inequality becomes an asymptotic equality, are characterized by their asymptotically uniform zero distributions. We also relate asymptotically extremal polynomials to the classical polynomials with asymptotically minimal norms.


References [Enhancements On Off] (What's this?)

  • 1. G. Aumann, Satz über das Verhalten von Polynomen auf Kontinuen, Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. (1933), 926-931.
  • 2. V. Avanissian and M. Mignotte, A variant of an inequality of Gel'fond and Mahler, Bull. London Math. Soc. 26 (1994), 64-68. MR 94j:32002
  • 3. B. Beauzamy, E. Bombieri, P. Enflo and H. L. Montgomery, Products of polynomials in many variables, J. Number Theory 36 (1990), 219-245.
  • 4. B. Beauzamy and P. Enflo, Estimations de produits de polynômes, J. Number Theory 21 (1985), 390-413. MR 91m:11015
  • 5. C. Benitez, Y. Sarantopoulos and A. Tonge, Lower bounds for norms of products of polynomials, Math. Proc. Cambridge Philos. Soc. 124 (1998), 395-408. MR 99h:46077
  • 6. H.-P. Blatt, E. B. Saff and M. Simkani, Jentzsch-Szego type theorems for the zeros of best approximants, J. London Math. Soc. 38 (1988), 307-316. MR 90a:30004
  • 7. P. B. Borwein, Exact inequalities for the norms of factors of polynomials, Can. J. Math. 46 (1994), 687-698. MR 95k:26015
  • 8. D. W. Boyd, Two sharp inequalities for the norm of a factor of a polynomial, Mathematika 39 (1992), 341-349. MR 94a:11162
  • 9. D. W. Boyd, Sharp inequalities for the product of polynomials, Bull. London Math. Soc. 26 (1994), 449-454. MR 95m:30008
  • 10. J. L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag, New York, 1984. MR 85k:31001
  • 11. G. Faber, Über Tschebyscheffsche Polynome, J. Reine Angew. Math. 150 (1920), 79-106.
  • 12. M. Fekete and J. L. Walsh, On the asymptotic behavior of polynomials with extremal properties, and of their zeros, J. Analyse Math. 4 (1955), 49-87. MR 17:354f
  • 13. A. O. Gel'fond, Transcendental and Algebraic Numbers, Dover, New York, 1960. MR 22:2598
  • 14. R. Grothmann, On the zeros of sequences of polynomials, J. Approx. Theory 61 (1990), 351-359. MR 91h:41006
  • 15. H. Kneser, Das Maximum des Produkts zweies Polynome, Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. (1934), 429-431.
  • 16. A. Kroó and I. E. Pritsker, A sharp version of Mahler's inequality for products of polynomials, Bull. London Math. Soc. 31 (1999), 269-278. MR 99m:30008
  • 17. N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972. MR 50:2520
  • 18. K. Mahler, An application of Jensen's formula to polynomials, Mathematika 7 (1960), 98-100. MR 23:A1779
  • 19. K. Mahler, On some inequalities for polynomials in several variables, J. London Math. Soc. 37 (1962), 341-344. MR 25:2036
  • 20. H. N. Mhaskar and E. B. Saff, The distribution of zeros of asymptotically extremal polynomials, J. Approx. Theory 65 (1991), 279-300. MR 92d:30005
  • 21. T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995. MR 96e:31001
  • 22. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Heidelberg, 1997. MR 99h:31001
  • 23. H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, New York, 1992. MR 93d:42029
  • 24. M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publ. Co., New York, 1975. MR 54:2990
  • 25. H. Widom, Polynomials associated with measures in the complex plane, J. Math. Mech. 16 (1967), 997-1014. MR 35:346
  • 26. H. Widom, Extremal polynomials associated with a system of curves in the complex plane, Adv. Math. 3 (1969), 127-232. MR 39:418

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 30C10, 11C08, 30C15, 31A05, 31A15

Retrieve articles in all journals with MSC (1991): 30C10, 11C08, 30C15, 31A05, 31A15


Additional Information

Igor E. Pritsker
Affiliation: Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, Oklahoma 74078-1058
Email: igor@math.okstate.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02856-2
Keywords: Polynomials, uniform norm, zero counting measure, logarithmic capacity, equilibrium measure, subharmonic function, Fekete points
Received by editor(s): December 27, 1997
Published electronically: May 21, 2001
Additional Notes: Research supported in part by the National Science Foundation grants DMS-9996410 and DMS-9707359.
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society