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.: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
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:74300 |
| Date | 31 March 2021 |
| Creators | Kley, Susanne |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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