On a Theorem by Bojanov and Naidenov applied to families of Gegenbauer-Sobolev polynomials (original) (raw)

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

Journal of Approximation Theory, 2010

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.

Sobolev extremal polynomials and applications

2020

Mención Internacional en el título de doctorLos polinomios ortogonales de Sobolev han sido estudiados en profundidad a lo largo de las últimas tres décadas como extensión natural de los polinomios ortogonales estándar. Existen tres grandes clasificaciones de los polinomios de Sobolev según las medidas involucradas en el producto interno que los origina: caso continuo, caso discreto, y el caso discreto-continuo. En nuestro trabajo se abordan cada uno de ellos en los Capítulos II, III y IV respectivamente.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: José Mauel Rodríguez García.- Secretario: Ramón Ángel Orive Rodríguez.- Vocal: Ana Pilar Foulquié Moren

Analytic aspects of Sobolev orthogonal polynomials revisited

Journal of Computational and Applied Mathematics, 2001

This paper surveys some recent achievements in the analytic theory of polynomials orthogonal with respect to inner products involving derivatives. Asymptotic properties, zero location, approximation and moment theory are some of the topic considered.

An extremal property of Hermite polynomials

Journal of Mathematical Analysis and Applications, 2004

Let H n be the nth Hermite polynomial, i.e., the nth orthogonal on R polynomial with respect to the weight w(x) = exp(−x 2 ). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f | |H n | at the zeros of H n+1 , then for k = 1, . . . , n we have f (k) H (k) n , where · is the L 2 (w; R) norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the L 2 (w; R) norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.

Some properties of zeros of Sobolev-type orthogonal polynomials

Journal of Computational and Applied Mathematics, 1996

For polynomials orthogonal with respect to a discrete Sobolev product, we prove that, for each n, Q n has at least n-m zeros on the convex hull of the support of the measure, where m denotes the number of terms in the discrete part. Interlacing properties of zeros are also described.

Concerning Asymptotic Behavior for Extremal Polynomials Associated to Nondiagonal Sobolev Norms

Journal of Function Spaces and Applications, 2013

Let ℙ be the space of polynomials with complex coefficients endowed with a nondiagonal Sobolev norm∥ · ∥W1,p(Vμ), where the matrixVand the measureμconstitute ap-admissible pair for1≤p≤∞. In this paper we establish the zero location and asymptotic behavior of extremal polynomials associated to∥ · ∥W1,p(Vμ), stating hypothesis on the matrixVrather than on the diagonal matrix appearing in its unitary factorization.