S--polyregular Bargmann spaces

Abstract

We introduce two classes of right quaternionic Hilbert spaces in the context of slice polyregular functions, generalizing the so-called slice and full hyperholomorphic Bargmann spaces. Their basic properties are discussed, the explicit formulas of their reproducing kernels are given and associated Segal--Bargmann transforms are also introduced and studied. The spectral description as special subspaces of $L^2$-eigenspaces of a second order differential operator involving the slice derivative is investigated.