Orthogonal polynomials and linear functionals. An algebraic approach and applications
Quoting the abstract of this short book in its entirety: ``This book presents an introduction to orthogonal polynomials, with an algebraic flavor, based on linear functionals defining the orthogonality and the Jacobi matrices associated with them. Basic properties of their zeros as well as quadrature rules are discussed. A key point is the analysis of those functionals satisfying Pearson equations (semiclassical case) and the hierarchy based on their class. The book's structure reflects the fact that its content is based on a set of lectures delivered by one of the authors at the first Orthonet Summer School in Seville, Spain in 2016. The presentation of the material is self-contained and will be valuable to students and researchers interested in a novel approach to the study of orthogonal polynomials, focusing on their analytic properties.'' This reviewer found that abstract to be quite accurate, although the book is very short for the amount of material that it touches on, making the claim of being ``self-contained'' a little iffy. The authors do point out ``Hoping to make up for this lack of exhaustiveness, we have added a list of references that an interested reader can consult.'' Their literature survey is not comprehensive either, but it does contain useful links. Indeed the book, short as it is, contains several ideas that were very surprising to me, and details of proofs that I had not seen elsewhere and which I found to be very interesting and valuable. First, the idea of using linear functionals to unify the treatment of orthogonal polynomials was new to me, although I have used orthogonal polynomials for, well, decades. I found the unity that this idea provides quite remarkable. Second, the book contains a very short but very illuminating chapter on potential theory, a classical subject for orthogonal polynomials that I had previously only seen in approximation theory (see e.g. Nick Trefethen's book Approximation Theory and Approximation Practice). The book is extremely well-written (my only cavil is the use of ``It is clear that'' is a bit too frequent, and indeed when the authors use it in Chapter 3 to claim that a general continued fraction converges, well, this is not even true, much less ``clear''. This error is only of small consequence, however, and is the only one I noticed). The book would make a nice text for an advanced short course in orthogonal polynomials; however, the instructors would have to provide their own exercises for the students.
scientific article; zbMATH DE number 7344232