- AutorIn
- Diplom-Mathematikerin Susanne Kley
- Titel
- The Truncated Matricial Hamburger Moment Problem and Corresponding Weyl Matrix Balls
- Zitierfähige Url:
- https://nbn-resolving.org/urn:nbn:de:bsz:15-qucosa2-743009
- Datum der Einreichung
- 13.09.2020
- Datum der Verteidigung
- 15.02.2021
- Abstract (EN)
- The present thesis intents on analysing the truncated matricial Hamburger power moment problem in the general (degenerate and non-degenerate) case. Initiated due to manifold lines of research, by this time, outnumbering results and thoughts have been established that are concerned with specific subproblems within this field. The resulting presence of such a diversity as well as an extensively considered topic si- multaneously involves advantageous as well as obstructive aspects: on the one hand, we adopt the favourable possibility to capitalise on essential available results that proved beneficial within subsequent research. Nevertheless, on the other hand, we are obliged to illustrate major preparatory work in order to illucidate the comprehension of the attaching examination. Moreover, treating the matricial cases of the respective problems requires meticulous technical demands, in particular, in view of the chosen explicit approach to solving the considered tasks. Consequently, the first part of this thesis is dedicated to furnishing the necessary basis arranging the prime results of this research paper. Compul- sary notation as well as objects are introduced and thoroughly explained. Furthermore, the required techniques in order to achieve the desired results are characterised and ex- haustively discussed. Concerning the respective findings, we are afforded the opportunity to seise presentations and results that are, by this time, elaborately studied. Being equipped with mandatory cognisance, the thematically bipartite second and pivo- tal part objectives to describe all the possible values of all the solution functions of the truncated matricial Hamburger power moment problem M P [R; (s j ) 2n j=0 , ≤]. Aming this, we realise a first paramount achievement epitomising one of the two parts of the main results: Capturing an established representation of the solution set R 0,q [Π + ; (s j ) 2n j=0 , ≤] of the assigned matricial Hamburger moment problem via operating a specific algorithm of Schur-type, we expand these findings. We formulate a parameterisation of the set R 0,q [Π + ; (s j ) 2n j=0 , ≤] which is compatible with establishing respective equivalence classes within a certain subset of Nevanlinna pairs and utilise specific systems of orthogonal polynomials in order to entrench novel representations. In conclusion, we receive a para- meterisation that is valid within the entire upper open complex half-plane Π + . The second of the two prime parts changes focus to analysing all possible values of the functions belonging to R 0,q [Π + ; (s j ) 2n j=0 , ≤] in an arbitrary point w ∈ Π + . We gain two decisive conclusions: We identify these respective values to exhaust particular matrix balls 2n K[(s j ) 2n j=0 , w] := {F (w) | F ∈ R 0,q [Π + ; (s j ) j=0 , ≤]} the parameters of which are feasable to being described by specific rational matrix-valued functions and, in this course, enhance formerly established analyses. Moreover, we compile an alternative representation of the semi-radii constructing the respective matrix balls which manifests supportive in further consideration. We seise the achieved parameterisation of the set K[(s j ) 2n j=0 , w] and examine the behaviour of the respective sequences of left and right semi-radii. We recognise that these sequences of semi-radii associated with the respective matrix balls in the general case admit a particular monotonic behaviour. Consequently, with increasing number of given data, the resulting matrix balls are identified as being nested. Moreover, a proper description of the limit case of an infinite number of prescribed moments is facilitated.
- Freie Schlagwörter (EN)
- moment problems, truncated Hamburger moment problem, Weyl matrix balls
- Klassifikation (DDC)
- 500
- Den akademischen Grad verleihende / prüfende Institution
- Universität Leipzig, Leipzig
- Version / Begutachtungsstatus
- angenommene Version / Postprint / Autorenversion
- URN Qucosa
- urn:nbn:de:bsz:15-qucosa2-743009
- Veröffentlichungsdatum Qucosa
- 31.03.2021
- Dokumenttyp
- Dissertation
- Sprache des Dokumentes
- Englisch
- Lizenz / Rechtehinweis
- CC BY 4.0
- Inhaltsverzeichnis
1. Brief Historic Embedding and Introduction 2. Part I: Initialising Compulsary Cognisance Arranging Principal Achievements 2.1. Notation and Preliminaries 2.2. Particular Classes of Holomorphic Matrix-Valued Functions 2.3. Nevanlinna Pairs 2.4. Block Hankel Matrices 2.5. A Schur-Type Algorithm for Sequences of Complex p × q Matrices 2.6. Specific Matrix Polynomials 3. Part II: Momentous Results and Exposition – Improved Parameterisations of the Set R 0,q [Π + ; (s j ) 2n j=0 , ≤] 3.1. An Essential Step to a Parameterisation of the Solution Set R 0,q [Π + ; (s j ) 2n j=0 , ≤] 3.2. Parameterisation of the Solution Set R 0,q [Π + ; (s j ) 2n j=0 3.3. Particular Matrix Polynomials 3.4. Description of the Solution Set of the Truncated Matricial Hamburger Moment Problem by a Certain System of Orthogonal Matrix Polynomials 4. Part III: Prime Results and Exposition – Novel Description Balls 4.1. Particular Rational Matrix-Valued Functions 4.2. Description of the Values of the Solutions 4.3. Monotony of the Semi-Radii and Limit Balls of the Weyl Matrix 5. Summary of Principal Achievements and Prospects A. Matrix Theory B. Integration Theory of Non-Negative Hermitian Measures