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- Musonda, John, 1981-
(författare)
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Reordering in Noncommutative Algebras, Orthogonal Polynomials and Operators
- 2018
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The main object studied in this thesis is the multi-parametric family of unital associative complex algebras generated by the element $Q$ and the finite or infinite set $\{S_j\}_{j\in J}$ of elements satisfying the commutation relations $S_jQ=\sigma_j(Q)S_j$, where $\sigma_j$ is a polynomial for all $j\in J$. A concrete representation is given by the operators $Q_x(f)(x)=xf(x)$ and $\alpha_{\sigma_j}(f)(x)=f(\sigma_j(x))$ acting on polynomials or other suitable functions. The main goal is to reorder arbitrary elements in this family and some of its generalizations, and to study properties of operators in some representing operator algebras, including their connections to orthogonal polynomials. For $J=\{1\}$ and $\sigma(x)=x+1$, the above commutation relations reduce to the famous classical Heisenberg--Lie commutation relation $SQ-QS=S$. Reordering an element in $S$ and $Q$ means to bring it, using the commutation relation, into a form where all elements $Q$ stand either to the left or to the right. For example, $SQ^2=Q^2S+2QS+S$. In general, one can use the commutation relation $SQ-QS=S$ successively and transform for any positive integer $n$ the element $SQ^n$ into a form where all elements $Q$ stand to the left. The coefficients which appear upon reordering in this case are the binomial coefficients. General reordering formulas for arbitrary elements in noncommutative algebras defined by commutation relations are important in many research directions, open problems and applications of the algebras and their operator representations. In investigation of the structure, representation theory and applications of noncommutative algebras, an important role is played by the explicit description of suitable normal forms for noncommutative expressions or functions of generators. Further investigation of the operator representations of the commutation relations by difference type operators on Hilbert function spaces leads to interesting connections to functional analysis and orthogonal polynomials. This thesis consists of two main parts. The first part is devoted to the multi-parametric family of algebras introduced above. General reordering formulas for arbitrary elements in this family are derived, generalizing some well-known results. As an example of an application of the formulas, centralizers and centers are computed. Some operator representations of the above algebras are also described, including considering them in the context of twisted derivations. The second part of this thesis is devoted to a special representation of these algebras by difference operators associated with action by shifts on the complex plane. It is shown that there are three systems of orthogonal polynomials of the class of Meixner--Pollaczek polynomials that are connected by these operators. Boundedness properties of two singular integral operators of convolution type connected to these difference operators are investigated in the Hilbert spaces related to these systems of orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on the $L^2$-spaces and estimates of the norms are obtained. This investigation is also extended to $L^p$-spaces on the real line where it is proved again that the two operators are bounded.
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| 2. |
- Musonda, John, et al.
(författare)
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Three systems of orthogonal polynomials and L2-boundedness of associated operators
- 2018
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 459:1, s. 464-475
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we describe three systems of orthogonal polynomials belonging to the class of Meixner-Pollaczek polynomials, and establish some useful connections between them in terms of three basic operators that are related to them. Furthermore, we investigate boundedness properties of two other operators, both as convolution operators in the translation invariant case where we use Fourier transforms and for the weights related to the relevant orthogonal polynomials. We consider only the most important but also simplest case of L-2-spaces. However, in subsequent papers, we intend to extend the study to L-p-spaces (1 < p < infinity).
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| 3. |
- Zulu, Joseph M., et al.
(författare)
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Community based interventions for strengthening adolescent sexual reproductive health and rights : how can they be integrated and sustained? A realist evaluation protocol from Zambia
- 2018
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Ingår i: Reproductive Health. - : BioMed Central. - 1742-4755. ; 15
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Tidskriftsartikel (refereegranskat)abstract
- Background: Research that explores how community-based interventions for strengthening adolescent sexual reproductive health and rights (SRHR) can be integrated and sustained in community health systems, is, to the best of our knowledge, very scarce, if not absent. It is important to document mechanisms that shape integration process in order to improve health systems' responsiveness towards adolescents' SRHR. This realist evaluation protocol will contribute to this knowledge in Zambia where there is increased attention towards promoting maternal, neonatal and child health as a means of addressing the current high early pregnancy and marriage rates. The protocol will ascertain: why, how, and under what conditions the integration of SRHR interventions into Zambian community health systems will optimise (or not) acceptability and adoption of SRHR services. This study is embedded within a randomized controlled trial - "Research Initiative to Support the Empowerment of Girls (RISE) "-which aims to reduce adolescent girl pregnancies and marriages through a package of interventions including economic support to families, payment of school fees to keep girls in school, pocket money for girls, as well as youth club and community meetings on reproductive health.Methods: This is a multiple-case study design. Data will be collected from schools, health facilities and communities through individual and group interviews, photovoice, documentary review, and observations. The study process will involve 1) developing an initial causal theory that proposes an explanation of how the integration of a community-based intervention that aimed to integrate adolescent SRHR into the community health system may lead to adolescent-friendly services; 2) refining the causal theory through case studies; 3) identifying contextual conditions and mechanisms that shape the integration process; and 4) finally proposing a refined causal theory and set of recommendations to guide policy makers, steer further research, and inform teaching programmes.Discussion: The study will document relevant values as well as less formal and horizontal mechanisms which shape the integration process of SRHR interventions at community level. Knowledge on mechanisms is essential for guiding development of strategies for effectively facilitating the integration process, scaling up processes and sustainability of interventions aimed at reducing SRH problems and health inequalities among adolescents.
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