  We consider a class of non-polynomial spline spaces over T-meshes, that is,
of spaces locally spanned both by polynomial and by suitably-chosen
non-polynomial functions, which we will refer to as generalized splines over
T-meshes. For such spaces, we provide, under certain conditions, a dimension
formula and a basis based on the notion of minimal determining set. We
explicitly examine some relevant cases, which enjoy a noteworthy behaviour with
respect to differentiation and integration; finally, we also study the
approximation power of the just constructed spline spaces.
