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

Aicke Hinrichs

ArticleOriginal scientific text

Title

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

Authors ¹

Affiliations

FSU Jena, Mathematisches Institut, 07740 Jena, Germany

Abstract

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^{- \frac{n}{2}} t_{Φ} (T) \leq c n^{- \frac{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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Title

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

Affiliations

Abstract

Bibliography