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On the type constants with respect to systems of characters of a compact abelian group

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We prove that there exists an absolute constant c such that for any positive integer n and any system Φ of $2^n$ characters of a compact abelian group, $2^{-n/2} t_Φ(T) ≤ c n^{-1/2} t_n(T)$, where T is an arbitrary operator between Banach spaces, $t_Φ(T)$ is the type norm of T with respect to Φ and $t_n(T)$ is the usual Rademacher type-2 norm computed with n vectors. For the system of the first $2^n$ Walsh functions this is even true with c=1. This result combined with known properties of such type norms provides easy access to quantitative versions of the fact that a nontrivial type of a Banach space implies finite cotype and nontrivial type with respect to the Walsh system or the trigonometric system.
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  • FSU Jena, Mathematisches Institut, 07740 Jena, Germany
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