Standard Setup Revisited

In the recent development of white noise theory the framework proposed by Cochran-KuO-Sengupta [4] has become more important for their characterization theorems, see also [1]. In fact, much attention has been paid to characterization theorems for the test functions $\mathcal{W} $ , for the generalized functions $\mathcal{W}^{*} $ , for white noise operators $\mathcal{L}(\mathcal{W}, \mathcal{W}^{*}) $ and for $\mathcal{L}(\mathcal{W}, \mathcal{W}) $. As was pointed out first by Chung-Chung-Ji [2], those characterization theorems are related each other however the statements are not unified because their objects are different so far as we are concerned with asingle CKS-space over aparticular underlying Gelfand triple. In this paper, using the standard setup of white noise calculus proposed by Hida-Obata-Sait\^o [6] and by Obata [12], we unify those characterization theorems into asingle statement. Let $S_{A}(U) $ and $S_{B}(V) $ be countable Hilbert nuclear spaces constructed from $L^{2}(U)$ and $L^{2}(V) $ in the standard manner, respectively, see 52. Then, for two weight sequences $\alpha=\{\alpha(n)\}_{n=0}^{\infty} $ and $\beta=\{\beta(n)\}_{n=0}^{\infty} $ we consider CKS-spaces of test white noise functions defined by $\mathcal{U}\equiv\Gamma_{\alpha}(S_{A}(U)) $ , $\mathcal{V}\equiv\Gamma_{\beta}(S_{B}(V)) $