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

Abstract

Abstract. Our object of study is the asymptotic behavior of the sequence of polynomials orthogonal with respect to the discrete Sobolev inner productwhere E is a rectifiabl Jordan curve or arc in the complex planeA is an M × M Hermitian matrix, Ml 1 + · · · + l m + m, |dξ | denotes the arc length measure, ρ is a nonnegative function on E, and z i ∈ , i = 1, 2, . . . , m, where is the exterior region to E.

Recurrence relations and outer relative asymptotics of orthogonal polynomials with respect to a discrete Sobolev type inner product

“…non-diagonal) Sobolev inner products with respect to measures supported on the complex plane can be found in [1,4,6,13]. Results related to asymptotics for extremal polynomials associated to non-diagonal Sobolev norms may be seen in [12,[18][19][20].…”

“…[9,10,16,[21][22][23]25]) nor from the point of view of Sobolev orthogonality, where it remains like a partially explored subject [3]. In fact, new results continue to appear in some recent publications [4,[6][7][8][11][12][13][14][15][18][19][20].…”

The Multiplication Operator, Zero Location and Asymptotic for Non-diagonal Sobolev Norms

“…Sobolev orthogonal polynomials on the unit circle and, more generally, on curves is a topic of recent and increasing interest in approximation theory; see, for instance, [4] and [11] (for the unit circle) and [23] and [2] (for the case of Jordan curves). The papers [1], [2], [4], [11], [20] and [22] deal with Sobolev spaces on curves and more general subsets of the complex plane.…”

“…The papers [1], [2], [4], [11], [20] and [22] deal with Sobolev spaces on curves and more general subsets of the complex plane.…”

“…The main aim of our paper is to obtain the same thesis as in Theorem 1.1, but with hypotheses directly on the matrix V (rather than on the diagonal matrix Λ which appears in its factorization); in exchange for a certain loss of generality, we require weaker hypothesis than (2).…”