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

Nobuhiro Asai·Izumi Kubo·Hui-Hsiung Kuo
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Abstract

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.

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Nobuhiro Asai, Izumi Kubo, Hui-Hsiung Kuo. 2001-04-12. Roles of Log-concavity, log-convexity, and growth order in white noise analysis. https://arxiv.org/abs/math/0104132

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