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Lacunary Müntz systems

  • Peter Borwein (a1) and Tamás Erdélyi (a2)
Abstract

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.

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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.

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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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