On the linear functionals associated to linearly related sequences of orthogonal polynomials

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:315

Article

Petronilho, J

关键词: orthogonal polynomials;    moment linear functionals;    inverse problems;    locally convex spaces;    Sobolev orthogonal polynomials;   

DOI  :  10.1016/j.jmaa.2005.05.018

来源: Elsevier

【 摘 要 】

An inverse problem is solved, by stating that the regular linear functionals u and v associated to linearly related sequences of monic orthogonal polynomials (P-n)(n) and (Q(n))(n), respectively, in the sense P-n(x) + Sigma(N)(i=1)r(i,n)P(n-i) (x) = Q(n)(x) + Sigma(M)(i=1)s(i,n)Q(n-i) (x) for all n = 0, 1, 2,... (where r(i,n) and s(i,n) are complex numbers satisfying some natural conditions), are connected by a rational modification, i.e., there exist polynomials phi and psi, with degrees M and N, respectively, such that phi u = psi v. We also make some remarks concerning the corresponding direct problem, stating a characterization theorem in the case N = 1 and M = 2. As an example, we give a linear relation of the above type involving Jacobi polynomials with distinct parameters. (c) 2005 Elsevier Inc. All rights reserved.

【 授权许可】

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