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 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 {α(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 application to the growth order of holomorphic functions on ℰc will also be discussed.
Let u be a positive continuous function on [0, ∞) satisfying the conditions:u is constructed by making use of the Legendre transform of u discussed in [4]. We prove a characterization theorem for generalized functions in [E] * u and also for test functions in [E]u in terms of their S-transforms under the same assumptions on u. Moreover, we give an intrinsic topology for the space [E]u of test functions and prove a characterization theorem for measures. We briefly mention the relationship between our method and a recent work by Gannoun et al. [10]. Finally, conditions for carrying out white noise operator theory and Wick products are given.
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