Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

Lacunary Müntz systems

Peter Borweina1 and Tamás Erdélyia2

a1 Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, Canada, B3H 3J5

a2 Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA

Abstract

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

S0013091500018472_eqnU1

is dense in C[0,1] in the uniform norm if and only if S0013091500018472_inline1. 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 xs2286 [0,1] and its Lebesgue measure µ(A) is at least ε > 0 then

S0013091500018472_eqnU2

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

(Received April 12 1991)