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

Abstract

Abstract. In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which the product of the function and its derivative appears. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials. The multiplication operator by the independent variable is the main tool in order to obtain our results.

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

“…[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].…”

“…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].…”

Zero location and asymptotic behavior for extremal polynomials with non-diagonal Sobolev norms

“…Thus, finding conditions to ensure the boundedness of M would provide important information about the crucial issue of determining the asymptotic behavior of Sobolev orthogonal polynomials. In this regard, some significant progress have been made in [2,11,5,16,19,17,18,22]. The more general result on this topic is [2,Theorem 8.1] which characterizes in terms of equivalent norms in Sobolev spaces the boundedness of M for the classical "diagonal" case ‖q‖ W k, p (µ 0 ,µ 1 ,...,µ N ) :=…”

“…The rest of the abovementioned papers provides conditions that ensure the equivalence of norms in Sobolev spaces, and consequently, the boundedness of M. For "non-diagonal" Sobolev norms, we can refer to [1,3,[7][8][9]11,14,16]. In particular, in [3,9,11,14,16] the authors study the asymptotic behavior of orthogonal polynomials with respect to non-diagonal Sobolev inner products. In [11] the authors deal with the asymptotic behavior of extremal polynomials with respect to the following non-diagonal Sobolev norms.…”

“…In [16] the authors analyze the boundedness of the zeros of the extremal polynomials for this norm under the ellipticity hypothesis |b| 2 ≤ (1 − ε)ac, µ-almost everywhere for some fixed 0 < ε ≤ 1.…”