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

Studia Mathematica (1996)

• Volume: 118, Issue: 3, page 231-243
• ISSN: 0039-3223

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## Abstract

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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, ${2}^{-n/2}{t}_{\Phi }\left(T\right)\le c{n}^{-1/2}{t}_{n}\left(T\right)$, where T is an arbitrary operator between Banach spaces, ${t}_{\Phi }\left(T\right)$ is the type norm of T with respect to Φ and ${t}_{n}\left(T\right)$ 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.

## How to cite

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Hinrichs, Aicke. "On the type constants with respect to systems of characters of a compact abelian group." Studia Mathematica 118.3 (1996): 231-243. <http://eudml.org/doc/216275>.

@article{Hinrichs1996,
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, $2^\{-n/2\} t_Φ(T) ≤ c n^\{-1/2\} t_n(T)$, 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.},
author = {Hinrichs, Aicke},
journal = {Studia Mathematica},
keywords = {characters; compact abelian group; Rademacher type-2 norm; finite cotype; nontrivial type; Walsh system; trigonometric system},
language = {eng},
number = {3},
pages = {231-243},
title = {On the type constants with respect to systems of characters of a compact abelian group},
url = {http://eudml.org/doc/216275},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Hinrichs, Aicke
TI - On the type constants with respect to systems of characters of a compact abelian group
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 3
SP - 231
EP - 243
AB - 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.
LA - eng
KW - characters; compact abelian group; Rademacher type-2 norm; finite cotype; nontrivial type; Walsh system; trigonometric system
UR - http://eudml.org/doc/216275
ER -

## References

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