Lp extremal polynomials. Results and perspectives

2000 Mathematics Subject Classification: 30C40, 30D50, 30E10, 30E15, 42C05.

Let α = β+γ be a positive finite measure defined on the Borel sets of C, with compact support, where β is a measure concentrated on a closed Jordan curve or on an arc (a circle or a segment) and γ is a discrete measure concentrated on an infinite number of points. In this survey paper, we present a synthesis on the asymptotic behaviour of orthogonal polynomials or Lp extremal polynomials associated to the measure α. We analyze some open problems and discuss new ideas related to their solving.