  In this paper we will develop a systematic method to answer the questions
$(Q1)(Q2)(Q3)(Q4)$ (stated in Section 1) with complete generality. As a result,
we can solve the difficulties $(D1)(D2)$ (discussed in Section 1) without
uncertainty. For these purposes we will introduce certain classes of growth
functions $u$ and apply the Legendre transform to obtain a sequence which leads
to the weight sequence $\{\a(n)\}$ first studied by Cochran et al. \cite{cks}.
The notion of (nearly) equivalent functions, (nearly) equivalent sequences and
dual Legendre functions will be defined in a very natural way. An application
to the growth order of holomorphic functions on $\ce_c$ will also be discussed.
