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In this paper we propose a local projector for truncated hierarchicalB-splines (THB-splines). The local THB-spline projector is an adaptation of theB\'ezier projector proposed by Thomas et al. (Comput Methods Appl Mech Eng 284,2015) for B-splines and analysis-suitable T-splines (AS T-splines). ForTHB-splines, there are elements on which the restrictions of THB-splines arelinearly dependent, contrary to B-splines and AS T-splines. Therefore, wecluster certain local mesh elements together such that the THB-splines withsupport over these clusters are linearly independent, and the B\'ezierprojector is adapted to use these clusters. We introduce general extensions forwhich optimal convergence is shown theoretically and numerically. In addition,a simple adaptive refinement scheme is introduced and compared to Giust et al.(Comput. Aided Geom. Des. 80, 2020), where we find that our simple approachshows promise.

A characterization of linear independence of THB-splines in $\mathbb{R}^n$ and application to B\'ezier projection