The AMS website will be down for maintenance on May 23 between 6:00am - 8:00am EDT. For questions please contact AMS Customer Service at or (800) 321-4267 (U.S. & Canada), (401) 455-4000 (Worldwide).


Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Generalizations of Müntz's Theorem via
a Remez-type inequality for Müntz spaces

Authors: Peter Borwein and Tamás Erdélyi
Journal: J. Amer. Math. Soc. 10 (1997), 327-349
MSC (1991): Primary 41A17; Secondary 30B10, 26D15
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The principal result of this paper is a Remez-type inequality for Müntz polynomials:

\begin{equation*}p(x) := \sum ^{n}_{i=0} a_{i} x^{\lambda _{i}}, \end{equation*}

or equivalently for Dirichlet sums:

\begin{equation*}P(t) := \sum ^{n}_{i=0}{a_{i} e^{-\lambda _{i} t}}\,,\end{equation*}

where $0 = \lambda _{0} < \lambda _{1} < \lambda _{2} <\cdots $. The most useful form of this inequality states that for every sequence $(\lambda _{i})^{\infty }_{i=0}$ satisfying $\sum ^{\infty }_{i=1} 1/\lambda _{i} < \infty $, there is a constant $c$ depending only on $\Lambda : = (\lambda _{i})^{\infty }_{i=0}$ and $s$ (and not on $n$, $\varrho $, or $A$) so that

\begin{equation*}\|p\|_{[0, \varrho ]} \leq c \,\|p\|_{A}\end{equation*}

for every Müntz polynomial $p$, as above, associated with $(\lambda _{i})^{\infty }_{i=0}$, and for every set $A \subset [\varrho ,1]$ of Lebesgue measure at least $s > 0$. Here $\|\cdot \|_{A}$ denotes the supremum norm on $A$. This Remez-type inequality allows us to resolve two reasonably long-standing conjectures.

The first conjecture it lets us resolve is due to D. J. Newman and dates from 1978. It asserts that if $\sum ^{\infty }_{i=1} 1/\lambda _{i} < \infty $, then the set of products $\{ p_{1} p_{2} : p_{1}, p_{2} \in \text {span} \{x^{\lambda _{0}}, x^{\lambda _{1}}, \ldots \}\} $ is not dense in $C[0,1]$.

The second is a complete extension of Müntz's classical theorem on the denseness of Müntz spaces in $C[0,1]$ to denseness in $C(A)$, where $A \subset [0,\infty )$
is an arbitrary compact set with positive Lebesgue measure. That is, for an
arbitrary compact set $A \subset [0,\infty )$ with positive Lebesgue measure,
$\text {span} \{ x^{\lambda _{0}}, x^{\lambda _{1}}, \ldots \} $ is dense in $C(A)$ if and only if $\sum ^{\infty }_{i=1} 1/\lambda _{i} =\infty $.

Several other interesting consequences are also presented.

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

  • 1. J. M. Anderson, Müntz-Szász type approximation and the angular growth of lacunary integral functions, Trans. Amer. Math. Soc. 169 (1972), 237-248. MR 46:9360
  • 2. J. Bak and D. J. Newman, Rational combinations of $x^{\lambda _{k}}, \lambda _{k} \geq 0$ are always dense in $C[0,1]$, J. Approx. Theory 23 (1978), 155-157. MR 58:6840
  • 3. S. N. Bernstein, Collected Works: Vol 1. Constructive Theory of Functions (1905-1930), English Translation, Atomic Energy Commission, Springfield, Va, 1958. MR 14:2c
  • 4. R. P. Boas, Entire Functions, Academic Press, New York, 1954. MR 16:914f
  • 5. P. B. Borwein, Zeros of Chebyshev polynomials in Markov Systems, J. Approx. Theory 63 (1990), 56-64. MR 92a:41002
  • 6. P. B. Borwein, Variations on Müntz's theme, Can. Math. Bull. 34 (1991), 305-310. MR 92i:41010
  • 7. P. B. Borwein and T. Erdélyi, Notes on lacunary Müntz polynomials, Israel J. Math. 76 (1991), 183-192. MR 93h:41014
  • 8. P. B. Borwein and T. Erdélyi, Lacunary Müntz systems, Proc. Edinburgh Math. Soc. 36 (1993), 361-374. MR 94i:41023
  • 9. P. B. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995. CMP 96:06
  • 10. P. B. Borwein, T. Erdélyi, and J. Zhang, Müntz systems and orthogonal Müntz polynomials, Trans. Amer. Math. Soc. 342 (1994), 523-542. MR 94f:42026
  • 11. E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966. MR 36:5568
  • 12. J. A. Clarkson and P. Erd\H{o}s, Approximation by polynomials, Duke Math. J. 10 (1943), 5-11. MR 4:196e
  • 13. T. Erdélyi, Remez-type inequalities on the size of generalized polynomials, J. London Math. Soc. 45 (1992), 255-264. MR 93e:41022
  • 14. T. Erdélyi, Remez-type inequalities and their applications, J. Comp. and Applied Math. 47 (1993), 167-210. MR 94m:26003
  • 15. G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1971.
  • 16. S. Karlin and W. J. Studden, Tchebycheff Systems with Applications in Analysis and Statistics, Wiley, New York, 1966. MR 34:4757
  • 17. W. A. J. Luxemburg and J. Korevaar, Entire functions and Müntz-Szász type approximation, Trans. Amer. Math. Soc. 157 (1971), 23-37. MR 43:7643
  • 18. C. Müntz, Über den Approximationsatz von Weierstrass, H. A. Schwartz Festschrift, Berlin, 1914.
  • 19. D. J. Newman, Derivative bounds for Müntz polynomials, J. Approx. Theory 18 (1976), 360-362. MR 55:3609
  • 20. D. J. Newman, Approximation with rational functions, vol. 41, Regional Conference Series in Mathematics, Providence, Rhode Island, 1979. MR 84k:41019
  • 21. G. Nürnberger, Approximation by Spline Functions, Springer-Verlag, Berlin, 1989. MR 90j:41025
  • 22. E. J. Remez, Sur une propriété des polynômes de Tchebyscheff, Comm. Inst. Sci. Kharkow 13 (1936), 93-95.
  • 23. T. J. Rivlin, Chebyshev Polynomials, 2nd ed., Wiley, New York, 1990. MR 92a:41016
  • 24. L. Schwartz, Etude des Sommes d'Exponentielles, Hermann, Paris, 1959. MR 21:5116
  • 25. P. W. Smith, An improvement theorem for Descartes systems, Proc. Amer. Math. Soc. 70 (1978), 26-30. MR 57:6985
  • 26. G. Somorjai, A Müntz-type problem for rational approximation, Acta. Math. Hung. 27 (1976), 197-199. MR 55:3622
  • 27. O. Szász, Über die Approximation steliger Funktionen durch lineare Aggregate von Potenzen, vol. 77, 1916, pp. 482-496.
  • 28. G. Szeg\H{o}, On the density of quotients of lacunary polynomials, Acta Math. Hung. 30 (1922), 149-154.
  • 29. A. K. Taslakyan, Some properties of Legendre quasi-polynomials with respect to a Müntz system, Mathematics 2 (1984), 179-189; Erevan University, Erevan. (Russian, Armenian Summary) MR 88e:33008
  • 30. M. von Golitschek, A short proof of Müntz Theorem, J. Approx. Theory 39 (1983), 394-395. MR 85b:41005

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 41A17, 30B10, 26D15

Retrieve articles in all journals with MSC (1991): 41A17, 30B10, 26D15

Additional Information

Peter Borwein
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Tamás Erdélyi
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Keywords: Remez inequality, M\"{u}ntz's Theorem, M\"{u}ntz spaces, Dirichlet sums, density
Received by editor(s): June 10, 1994
Received by editor(s) in revised form: September 20, 1996
Additional Notes: Research of the first author was supported, in part, by NSERC of Canada. Research of the second author was supported, in part, by NSF under Grant No. DMS-9024901 and conducted while an NSERC International Fellow at Simon Fraser University.
Dedicated: Dedicated to the memory of Paul Erdős
Article copyright: © Copyright 1997 by the authors

American Mathematical Society