Harmonic polynomials on the white noise space (Q1282070)

Let \(P_{\text{harm}}\) denote the space of harmonic polynomials of \(\mathbb{R}^{\mathbb{N}}\). The author investigates certain properties of \(P_{\text{harm}}\) on the white noise space. Specifically, it is shown that harmonic polynomials are characterized by an invariance property with respect to the \(S\)-transform, and that the closure \(\overline{P}_{\text{harm}}^{(S)}\) in Hida's test-functionals topology is exactly the kernel of the Gross Laplacian \(\Delta_G\). Furthermore, it is shown that \(\text{Ker } \Delta_G\) is strictly contained in the space of functionals of \((S)\) which remain fixed under \(S\)-transform.