Cesaro Hilbert space and the Levy laplacian

In this paper we introduce a new scalar product on distribution spaces based on the Cesaro mean of a sequence. We then use this scalar product to construct a family of separable Hilbert spaces $\mathcal{H}_C$, called Cesaro Hilbert spaces and naturally associated to the Levy Laplacian. Finally we use the essentially infinite dimensional character of the Levy Laplacian to construct a class of solutions of the Levy heat equation which has no finite dimensional (or ``regular'' infinite dimensional) analogue.