Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on a Rectifiable Jordan Curve or Arc

Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner product langle f, g rangle = ∈t_{E} f(ξ) overline{g(ξ)} ρ(ξ) |d ξ|+ f(Z) A g(Z)^H, where E is a rectifiable Jordan curve or arc in the complex plane f(Z) = (f(z_1), ldots, f^{(l_1)}(z_1) , ldots , f(z_m) , ldots ,f^{(l_m)}(z_m)), A is an M times M Hermitian matrix, M l ₁ + ⋅s + l _m + m , |d ξ| denotes the arc length measure, ρ is a nonnegative function on E , and z _i ∈Ω, i=1,2,ldots,m , where Ω is the exterior region to E . July 23, 1999. Dates revised: September 11, 2000 and February 16, 2001. Date accepted: February 26, 2001.