  A class of growth functions $u$ is introduced to construct Hida distributions
and test functions. The Legendre transform $\ell_{u}$ of $u$ is used to define
a sequence $\a(n)=(\ell_{u}(n) n!)^{-1}, n\geq 0$, of positive numbers. From
this sequence we get a CKS-space. Under various conditions on $u$ we show that
the associated sequence $\{\a(n)\}$ satisfies those conditions for carrying out
the white noise distribution theory on the CKS-space. We show that $u$ and its
dual Legendre transform $u^{*}$ are growth functions for test and generalized
functions, respectively, in the characterization theorems.
