Andrei Martinez-Finkelshtein

Multi-applications of multi-orthogonality

Orthogonal polynomials (OP) belong in a toolbox of any branch of modern science. They appear in multiple applications in engineering, physics, computer science and mathematics. One of the secrets of their ubiquity is that they are on the crossroads of several mathematical branches: analysis, approximation theory, spectral theory, special functions, combinatorics, mathematical physics, numerical analysis, to mention a few. OP have also stimulated the development of new tools and approaches to their study.

Their cousins, multiple orthogonal polynomials (MOP), defined by a combination of several orthogonality conditions, are much less known. Their origins can be traced back to the works of Hermite in number theory. However, a general theory of MOP started to develop basically less than 40 years ago. As it often happens, this was simultaneous to appearance of new applications, which brought up new challenges requiring new tools, some of them borrowed from other branches of analysis.

In this talk I will try to illustrate some parallels and some striking differences between OP and MOP, as well as to mention some possible emerging applications of MOP.