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Lacunary Müntz systems

Published online by Cambridge University Press:  20 January 2009

Peter Borwein  and
Tamás Erdélyi
Peter Borwein
Affiliation:
Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 3J5
Tamás Erdélyi
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA
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Abstract

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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+1/λi>1.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 3 , October 1993 , pp. 361 - 374
Copyright
Copyright © Edinburgh Mathematical Society 1993

References

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