International Mathematical Forum, 2, 2007, no. 39, 1927 - 1934 Extremal Polynomials with Varying Measures Rabah Khaldi Department of Mathematics, Annaba University B.P. 12, 23000 Annaba, Algeria rkhadi@yahoo.fr Fateh Aggoune Guelam University, Faculty of Sciences B.P 401, 24000 Guelma, Algeria faggoune@yahoo.fr Abstract We investigate the strong asymptotics for L p -extremal polynomials with respect to varying measures on a rectifiable Jordan curve perturbed by a finite Blaschke sequence of point masses outside the curve Mathematics Subject Classification: 42C05, 30E10 Keywords: Extremal Polynomials, Varying measures 1 Introduction Let σ be a finite positive Borel measure on a compact set of the complex plane whose support contains an infinite set of points. We denote by T n,2 (z)= z n + ...., the monic polynomial of degree n orthogonal with respect to the measure σ. One of the most useful tools in the study of orthogonal polynomials is the fact that they solve the following extremal problem: minimize L 2 (σ) norm for all monic polynonials of degree n i.e. ‖T n,2 ‖ 2 L 2 (σ) := min Q∈P n-1 ‖z n + Q‖ 2 L 2 (σ) = m n,2 (σ) where P n denotes the set of polynomials of degree at most n. This characterization of orthogonal polynomials permits us to define a larger class of polynomials called extremal polynomials that solve the extremal problem: