Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 2
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Erdélyi, Tamás and Johnson, William B. 2001. The “Full Müntz Theorem” inL p[0, 1] for 0<p<∞. Journal d'Analyse Mathématique, Vol. 84, Issue. 1, p. 145.

    Erdélyi, Tamás 1993. Remez-type inequalities and their applications. Journal of Computational and Applied Mathematics, Vol. 47, Issue. 2, p. 167.

  • Proceedings of the Edinburgh Mathematical Society, Volume 36, Issue 3
  • October 1993, pp. 361-374

Lacunary Müntz systems

  • Peter Borwein (a1) and Tamás Erdélyi (a2)
  • DOI:
  • Published online: 01 January 2009

The classical theorem of Müntz and Szász says that the span of

is dense in C[0,1] in the uniform norm if and only if . We prove that, if {λi} is lacunary, we can replace the underlying interval [0,1] by any set of positive measure. The key to the proof is the establishment of a bounded Remez-type inequality for lacunary Müntz systems. Namely if A ⊆ [0,1] and its Lebesgue measure µ(A) is at least ε > 0 then

where c depends only on ε and Λ (not on n and A) and where Λ:=infiλi+1i>1.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

1.P. B. Borwein , Zeros of Chebyshev polynomials in Markov systems, J. Approx. Theory 63 (1990), 5664.

2.P. B. Borwein and E. Saff , On the denseness of weighted incomplete approximations, in A. A. Gonchar and E. B. Saff (Eds.), Progress in Approximation Theory (Springer-Verlag, 1992), 419429.

3.P. B. Borwein , Variations on Müntz's theme, Canad. Math. Bull. 34 (1991), 305310.

5.J. A. Clarkson and P. Erdös , Approximation by polynomials, Duke Math J. 10 (1943), 511.

8.M. Von Golitschek , A short proof of Müntz's theorem, J. Approx. Theory 39 (1983), 394395.

9.I. I. Hirschman Jr., Approximation by non-dense sets of functions, Ann. of Math. 50 (1949), 666675.

12.D. J. Newman , Derivative bounds for Müntz polynomials, J. Approx. Theory 18 (1976), 360362.

15.P. W. Smith , An improvement theorem for Descartes systems, Proc. Amer. Math. Soc. 70 (1978), 2630.

16.T. T. Trent , A Müntz-Szász theorem for C(D), Proc. Amer. Math. Soc. 83 (1981), 296298.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *