Cardinal Hermite exponential spline functions are a generalization of the classical cardinal Hermite polynomial splines. In this work we consider the 4-dimensional space $ ε _{ 4 } $ = {1, x, $ e ^{ \alpha x } $ , $ e ^{ -\alpha x } $ } with α ∈ $ ℝ ^{ + } $ ∪ i $ ℝ ^{ + } $ , and therefore a generalization of the well-known cubic cardinal Hermite polynomial splines. For this class of Hermite spline functions, here denoted by $ ε _{ 4 } $ -Hermite splines, we establish the connection to standard exponential splines, we show stability and approximation power, and we emphasize their capability of reproducing elliptical and circular shapes. Finally, we investigate their multiresolution properties and we propose a non-stationary Hermite interpolatory subdivision scheme for refinement of vector sequences via the repeated application of level-dependent matrix subdivision operators.