  In this paper hierarchical analysis-suitable T-splines (HASTS) are developed.
The resulting spaces are a superset of both analysis-suitable T-splines and
hierarchical B-splines. The additional flexibility provided by the hierarchy of
T-spline spaces results in simple, highly localized refinement algorithms which
can be utilized in a design or analysis context. A detailed theoretical
formulation is presented including a proof of local linear independence for
analysis-suitable T-splines, a requisite theoretical ingredient for HASTS.
B\'{e}zier extraction is extended to HASTS simplifying the implementation of
HASTS in existing finite element codes. The behavior of a simple HASTS
refinement algorithm is compared to the local refinement algorithm for
analysis-suitable T-splines demonstrating the superior efficiency and locality
of the HASTS algorithm. Finally, HASTS are utilized as a basis for adaptive
isogeometric analysis.
