Sobolev extremal polynomials and applications

• Autores: Abel Díaz González
• Directores de la Tesis: Héctor Esteban Pijeira Cabrera (dir. tes.)
• Lectura: En la Universidad Carlos III de Madrid ( España ) en 2020
• Idioma: español
• Tribunal Calificador de la Tesis: José Manuel Rodríguez García (presid.), Ramón Angel Orive Rodríguez (secret.), Ana Pilar Foulquié Moreno (voc.)
• Programa de doctorado: Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de Madrid
• Materias:
  • Matemáticas
    • Análisis y análisis funcional
      • Teoría de la aproximación
      • Funciones de una variable compleja
      • Teoría de potencial
      • Funciones especiales
• Enlaces
  • Tesis en acceso abierto en: e-Archivo
• Resumen

  • The aim of this doctoral dissertation is the study of certain analytic and algebraic properties of polynomials extremal with respect to a Sobolev p-norm and its applications. Problems related to the three cases of Sobolev norms (continuous, discrete and discrete-continuous) are studied independently.

    In the Continuous case, the extremal polynomials with respect to Sobolev norms of type p are characterized by means of an operator. Results on the location of the zeros of these polynomials as well as its asymptotic behavior are obtained.

    For extremal polynomials with respect to a discrete Sobolev norm, generated by an inner product, we give a sufficient condition under which the zeros of these polynomials are real and simple The classical Markov's theorem of rational approximation is extended this case of the Sobolev-type orthogonality.

    In the Discrete-continuous case, the Jacobi-Sobolev space is studied and its completeness is characterized. It is described the cases when the Fourier series converge in Sobolev p-norm.