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

Abstract

In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a non-diagonal Sobolev norm in the worst case, i.e., when the quadratic form is allowed to degenerate. 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.

Asymptotics for Laguerre–Sobolev type orthogonal polynomials modified within their oscillatory regime

“…non-diagonal) Sobolev inner products with respect to measures supported on the complex plane can be found in [1,4,7,28]. On the other hand, results concerning asymptotics for extremal polynomials associated to non-diagonal Sobolev norms may be seen in [29,[33][34][35].…”

Measurable diagonalization of positive definite matrices

“…[7][8][9][10][11] and references therein). It is possible that the first result in the literature about asymptotic properties for orthogonal polynomials with respect to a non-discrete Sobolev type inner product associated to general measures is contained in [8], where the authors show how to obtain the nth root asymptotic of Sobolev orthogonal polynomials if the zeros of these polynomials are contained in a compact set of the complex plane.…”

Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms

“…Although it is required compact support for , this is, certainly, a natural hypothesis: if ( ) is not bounded, then we cannot expect to have zeros uniformly bounded, not even in the classical case (orthogonal polynomials in 2 ); see [21]. Taking = 1, 1 ≤ ≤ 2 and setting up hypothesis on the matrix (see (4)) rather than on the diagonal matrix , the authors of [22] the following equivalent result to [5,Theorem 1].…”