Ideal norms and trigonometric orthonormal systems
  In this article, we characterize the $UMD$--property of a Banach space $X$ by ideal norms associated with trigonometric orthonormal systems.   The asymptotic behavior of that numerical parameters can be used to decide whether or not $X$ is a $UMD$--space. Moreover, in the negative case, we obtain a measure that shows how far $X$ is from being a $UMD$--space.   The main result is, that all described parameters are equivalent also in the quantitative setting. 
Joerg Wenzel