Matrix bispectrality and noncommutative algebras: beyond the prolate spheroidals. (English) Zbl 1543.81134
Summary: 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.
MSC:
| 81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |
| 35C08 | Soliton solutions |
| 15A30 | Algebraic systems of matrices |
| 47A10 | Spectrum, resolvent |
| 13N10 | Commutative rings of differential operators and their modules |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
Keywords:
bispectral algebras; bispectral triple; presentations of finitely generated algebras; time-and-band limiting; integral and commuting differential operatorsReferences:
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