Orthogonal Polynomials
Project Summary
| Period | 2009-01-01 to Present |
|---|
Exceptional orthogonal polynomials came as a little surprise to the community of mathematicians working on special functions and approximation theory. Classical orthogonal polynomials are known since the mid XIX century, and they appear in numerous applications in physics (optics, quantum mechanics, electromagnetism, stochastic processes), engineering (signal processing, compression), numerical methods (Galerkin methods, FEM, etc.) and Machine Learning (PINNs, kernel methods, KANs, dimensionality reduction, etc.).
Classical orthogonal are, so to say, the building bricks one uses to write and approximate many complex functions.One of the main reasons for this ubiquitous presence is the fact that they constitute polynomial bases of Sturm-Liouville problems, i.e. they span a complete Hilbert space of polynomial eigenfunctions to a second order differential eigenvalue problem with suitable boundary conditions.
Since the work of Solomon Bochner, almost 100 years ago, it was believed that the orthogonal polynomial families of Hermite, Laguerre and Jacobi (including all the subcases like Chebyshev, Legendre, etc.) were the only polynomial bases of Sturm-Liouville problems, and indeed this was often quoted a way to characterize and single out classical orthogonal polynomials. Around 2008, working with my colleagues Niky Kamran at McGill and Robert Milson at Dalhousie, we discovered that the class of polynomial Sturm-Liouville problems could be enlarged, provided one was ready to relax one typical assumption mathematicians always took for granted: that the infinite basis contains a polynomial for every degree.
We originally coined the name exceptional orthogonal polynomials for this new class because we believed that they were very rare, and indeed we first found them via rather involved group theoretic arguments. Soon after we published our article, Christianne Quesne (ULB) showed that our new examples could be obtained by applying Darboux transformations to classical families, and this was a seminal discovery, since iterated Darboux transformations were the key tool to generate larger families of exceptional orthogonal polynomials, as initiated shortly after by Odake and Sasaki.
Together with many collaborators, but especially with my friend Rob Milson, I have devoted the past 20 years of my life to developing the theory of exceptional orthogonal polynomials, which has also attracted many talented researchers worldwide. We are now close to providing a complete classification of all such families, in much the same way as Bochner did in 1929. Yet many of the novel applications that these new building blocks unravel are yet to be developed.
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