CKS-space in terms of growth functions

TL;DR

This paper introduces a new class of growth functions to construct and analyze CKS-spaces within white noise distribution theory, utilizing Legendre transforms to define sequences that characterize test and generalized functions.

Contribution

It develops a framework linking growth functions and Legendre transforms to construct CKS-spaces suitable for white noise analysis, expanding the theoretical foundation.

Findings

Established conditions for growth functions to define CKS-spaces

Connected growth functions with test and generalized functions via Legendre transforms

Provided characterization theorems for white noise distribution spaces

Abstract

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.