On Sobolev orthogonal polynomials

Sobolev orthogonal polynomials have been studied extensively in the past quarter-century. The research in this field has sprawled into several directions and generates a plethora of publications. This paper contains a survey of the main developments up to now. The goal is to identify main ideas and developments in the field, which hopefully will lend a structure to the mountainous publications and help future research.The project was carried out when the second author was on sabbatical from the University of Oregon and visited the Carlos III University of Madrid under its generous Excellence Chair Program. The work of the first author has been supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, Grant MTM2012-36732-C03-01. The work of the second author was supported in part by National Science Foundation Grant DMS-1106113