Roles of log-concavity, log-convexity, and growth order in white noise analysis

In this paper we will develop a systematic method to answer the questions ( Q1) (Q2) (Q3) (Q4) (stated in Sec. 1) with complete generality. As a result , we can solve the difficulties (D1) (D2) (discussed in Sec. 1) without unc ertainty. For these purposes we will introduce certain classes of growth fu nctions u and apply the Legendre transform to obtain a sequence which leads to the weight sequence {alpha (n)} first studied by Cochran et al.(6) The notion of (nearly) equivalent functions, (nearly) equivalent sequences and dual Legendre functions will be defined in a very natural way, An applicati on to the growth order of holomorphic functions on epsilon (c) will also be discussed.