Analytic properties of polynomials orthogonal with respect to coherent pair of measures supported on the unit circle

• Autores: Lino Gustavo Garza Gaona
• Directores de la Tesis: Francisco Marcellán Español (dir. tes.) , Luis Enrique Garza Gaona (codir. tes.)
• Lectura: En la Universidad Carlos III de Madrid ( España ) en 2017
• Idioma: español
• Tribunal Calificador de la Tesis: Guillermo Tomás López Lagomasino (presid.) , Amilcar Jose Pinto Lopes Branquinho (secret.) , María Alicia Cachafeiro López (voc.)
• Enlaces
  • Tesis en acceso abierto en: e-Archivo
• Resumen

  • This work presents a study of orthogonal polynomials from a matrix point of view. We deal with semiclassical and coherent orthogonal polynomials. We also consider the difference and q-difference operators. We also study the coherent pairs of measures supported on the unit circle.

    In Chapter 1 we present the framework on which the different chapters are developed as well as an historical overview about coherent pairs on the real line and on the unit circle, where it should be noted that there exist only few works and it is a fi eld yet to explore.

    Besides, in Chapter 2 a new tool introduced by L. Verde-Star in [61] and [62], based on the representation of a sequence of polynomials {P_n(x)}_{n≥0} by using an in nite lower semi-matrix, is presented in a very detailed way. This tool is used to characterize sequences of classical orthogonal polynomials and sequences of discrete and classical q-orthogonal polynomials in terms of banded matrices. To do this, the polynomials are represented with respect to the monomial basis {x^k}_{k≥0} as P_k(x) = ∑_{j=0}^{k} a_{k,j} x^{j} ; a_{k,k} = 1, k≥0, and the coefficients a_{k,j} are used as entries of the matrix A = [a_{k,j} ]. Several properties of orthogonal polynomials can be easily obtained with this approach.

    Afterwards, in Chapter 3, a new characterization of semiclassical orthogonal polynomials is presented by using the matrix approach described in Chapter 2. Semiclassical orthogonal polynomials are usually de ned by a structure relation such as ø(x)P_{n}^{[1]}(x) = ∑_{k=n-s}^{n+t} a_{n,k}P_k(x); n≥s; a_{n,n-s}≠0, n≥s+1, where ø is a polynomial of degree t and fP[1]n (x)gn0 denotes the monic sequence of the first derivatives P_{n}^{[1}] (x) = [P_{n+1}(x)]/(n+1).

    This is the so-called first structure relation and it was introduced by P. Maroni ([49]). Besides, we give a relation between the Jacobi matrix associated with semiclassical orthonormal polynomials and the matrix whose entries are the coefficients appearing in the previous structure relation. We also use the matrix approach to characterize when a pair of sequences of orthogonal polynomials constitutes a coherent pair. The most general characterization for the coherence on the real line was given by M. N. de Jesús, F. Marcellán, J. Petronilho, and N. C. Pinzón-Cortés ([20]). They studied the structure relation ∑_{i=0}^{M} c_{i,n}P_{n+m-i}^{[m]}(x) =∑_{i=0}^{N} b_{i,n}Q_{n+k-i}^{[k]}(x), n≥0, where, M,N, m, k are non negative integers and the constants {c_{i,n}}, {b_{i,n}} satisfy some natural conditions. By using the matrix approach, the coherence relation is equivalent to the existence of a banded matrix whose size is related to the class of coherence. We have studied the cases of (1,0)-coherence and (M,0)-coherence which are characterized with a (0,1)-banded matrix and a (0,M)-banded matrix, respectively. In the same way, we have studied the cases of (1,0)-coherence of order m and (M,0)-coherence of order m, where the order stands for the quantity of derivatives taken over the terms of the sequence of orthogonal polynomials corresponding to the measure 0. In this situation we obtained similar results as in the case of no derivatives. It is worth mentioning that in both cases we carry out numerical experiments to illustrate the results we developed.

    In Chapter 4, we consider the action of the difference operator and the divided difference operator D in the linear space of polynomials with real coefficients. D is either Dw or Dq defined by (Dwp)(x) = [p(x + w) - p(x)]/w; for w in C \{0}, (Dqp)(x) = [p(qx) - p(x)]/[(q - 1)x], for x ≠ 0; (Dqp)(0) = p'(0); q in C \ {0}.

    We obtain a new structure relation for the Dv-semiclassical orthogonal polynomials that is used later on to deduce the Dv-semiclassical character of a given linear functional in terms of banded matrices. We also developed a similar characterization for the Dv-coherence.

    Finally, in Chapter 5 we present a study of polynomials which are orthogonal with respect to measures supported on the unit circle, having as a goal to give a characterization of all the pairs of Borel positive measures (µ_0, µ_1) supported on the unit circle such that the corresponding sequences of monic orthogonal polynomials {ø_n(µ_0; z)}_{n≥0} and {ø_n(µ_1; z)}_{n≠0} satisfy the following relation 1/(n + 1) ø'_{n+1}(µ_0; z) = ø_n(µ_1; z) + a_nø_{n-1}(µ_1; z) + b_nø_{n-2}(µ_1; z), n≥3.

    This problem is an extension of the one studied in [45] by F. Marcellan and A. Sri Ranga where they consider one term less in the right hand side of the previous equation. This work is contained in Chapter 5.