Spectral problems for generalized Jacobi matrices

• Published in: Linear algebra and its applications Vol. 382; pp. 1 - 24
• Main Authors: Derevyagin, Maxim, Derkach, Vladimir
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01-05-2004; Elsevier Science
• Subjects: Algebra; Continued fraction; Exact sciences and technology; Inverse spectral problem; Jacobi matrix; Linear and multilinear algebra, matrix theory; m-Function; Mathematics; Pade approximant; Schur algorithm; Sciences and techniques of general use; Secondary 30B70; Inverse spectral problem; 47A57; 41A21; 30E05; Primary 47B36; 47B50; Jacobi matrix; m-Function; Schur algorithm; Pade approximant; Continued fraction; Real function; Generalized function; Moment problem; Schur complement; Rational function; Generalized
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2003.11.003

A new class of generalized Jacobi matrices is introduced. Every proper real rational function is proved to be the
m-function of a unique finite generalized Jacobi matrix. Moreover, every generalized Nevanlinna function
m(·) which is a solution of a determinate indefinite moment problem turns out to be the
m-function of a unique infinite generalized Jacobi matrix. The method we use is based on the step-by-step Schur process of solving the indefinite moment problem. The convergence of the sequence of subdiagonal Pade approximants for the corresponding Hamburger series is investigated.