  The bispectral problem is motivated by an effort to understand and extend a
remarkable phenomenon in Fourier analysis on the real line: the operator of
time-and-band limiting is an integral operator admitting a second-order
differential operator with a simple spectrum in its commutator. In this
article, we discuss a noncommutative version of the bispectral problem,
obtained by allowing all objects in the original formulation to be
matrix-valued. Deep attention is given to bispectral algebras and their
presentations as a tool to get information about bispectral triples.
