Author:
Dubovoy Vladimir K.,Fritzsche Bernd,Kirstein Bernd,Mädler Conrad,Müller Karsten
Abstract
AbstractThis paper is a generalization of the topic handled in Bogner et al. (Oper Theory 1(1):55–95, 2007a, Oper Theory 1(2):235–278, 2007b) where the Schur–Potapov algorithm (SP-algorithm) was handled in the context of non-degenerate $${p\times q}$$
p
×
q
Schur sequences and non-degenerate $${p\times q}$$
p
×
q
Schur functions. In particular, the interplay between both types of algorithms was intensively studied there. This was itself a generalization of the classical Schur algorithm (Schur in J Reine Angew Math 148:122–145, 1918) to the non-degenerate matrix case. In treating the matrix case a result due to Potapov (Potapov in Trudy Moskov Mat Obšč 4:125–236, 1955) concerning particular linear fractional transformations of contractive $${p\times q}$$
p
×
q
matrices was used. For this reason, the notation SP-algorithm was already chosen in Dubovoj et al. (Matricial version of the classical Schur problem, volume 129 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1992). We are going to introduce both types of SP-algorithms as well for arbitrary $${p\times q}$$
p
×
q
Schur sequences as for arbitrary $${p\times q}$$
p
×
q
Schur functions. Again we will intensively discuss the interplay between both types of algorithms. Applying the SP-algorithm, a complete treatment of the matricial Schur problem in the most general case is established. A one-step extension problem for finite $${p\times q}$$
p
×
q
Schur sequences is considered. Central $${p\times q}$$
p
×
q
Schur sequences are studied under the view of SP-parameters.
Funder
Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany
Universität Leipzig
Publisher
Springer Science and Business Media LLC
Reference30 articles.
1. Arov, D.Z., Kreĭn, M.G.: The problem of finding the minimum entropy in indeterminate problems of continuation. Funktsional. Anal. i Prilozhen. 15(2), 61–64 (1981)
2. Arov, D.Z., Kreĭn, M.G.: Calculation of entropy functionals and their minima in indeterminate continuation problems. Acta Sci. Math. (Szeged) 45(1–4), 33–50 (1983)
3. Ball, J.A., Gohberg, I., Rodman, L.: Interpolation of Rational Matrix Functions. Operator Theory: Advances and Applications, vol. 45. Birkhäuser Verlag, Basel (1990)
4. Ben-Israel, A., Greville, T.N.E.: Generalized Inverses, volume 15 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 2nd edn. Springer, New York (2003). (Theory and applications)
5. Birkigt, S.: Ein matrizielles finites Potenzmomentenproblem vom Hamburger-Typ. Diplomarbeit, Universität Leipzig (2013)