Fast polynomial transforms

Classical orthogonal polynomials and the related associated functions are real classics in approximation theory. They share a rich history of research that has uncovered their many relationships to topics of fundamental importance. This text develops a new aspect of the so-called connection problem. This problem asks how a given expansion in a specific sequence of polynomials or functions may be converted into an equivalent one using a different sequence --- often within reason, that is, within the same classical family. A new theory relates this problem to the class of semiseparable matrices. This implies efficient algorithms that have the capacity to cover the connection problem not only numerically efficient, but at the same time, numerically stable. The result has implications for numerical problems whose treatment involves these transformations. One such example, described in more detail, are generalizations of the fast Fourier transform to geometries like the two-sphere or the rotation group SO(3).