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| 2. |
- Kaijser, Sten, et al.
(author)
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Lp-Boundedness of Two Singular Integral Operators of Convolution Type
- 2016
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In: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056
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Book chapter (peer-reviewed)abstract
- We investigate boundedness properties of two singular integral operators defined on Lp-spaces (1 < p < ∞) on the real line, both as convolution operators on Lp(R) and on the spaces Lp(w), where w(x) = 1/2cosh πx/2. It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for p = 2 and weak boundedness for p = 1, and then using interpolation to obtain boundedness for 1 < p ≤ 2. To obtain boundedness also for 2 ≤ p < ∞, we use duality in the translation invariant case, while the weighted case is partly based on the expositions on the conjugate function operator in [7].π/2
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| 3. |
- Musonda, John
(author)
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Orthogonal Polynomials, Operators and Commutation Relations
- 2017
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Licentiate thesis (other academic/artistic)abstract
- Orthogonal polynomials, operators and commutation relations appear in many areas of mathematics, physics and engineering where they play a vital role. For instance, orthogonal functions in general are central to the development of Fourier series and wavelets which are essential to signal processing. In particular, as demonstrated in this thesis, orthogonal polynomials can be used to establish the L2-boundedness of singular integral operators which is a fundamental problem in harmonic analysis and a subject of extensive investigations. The Lp-convergence of Fourier series is closely related to the Lp-boundedness of singular integral operators. Many important relations in physical sciences are represented by operators satisfying various commutation relations. Such commutation relations play key roles in such areas as quantum mechanics, wavelet analysis, representation theory, spectral theory, and many others.This thesis consists of three main parts. The first part presents a new system of orthogonal polynomials, and establishes its relation to the previously studied systems in the class of Meixner–Pollaczek polynomials. Boundedness properties of two singular integral operators of convolution type are investigated in the Hilbert spaces related to the relevant 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 L2-spaces, and estimates of the norms are obtained.The second part extends the investigation of the boundedness properties of the two singular integral operators to Lp-spaces on the real line, both in the weighted and unweighted spaces. It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by first proving boundedness for L2 and weak boundedness for L1, and then using interpolation to obtain boundedness for the intermediate spaces. To obtain boundedness for the remaining spaces, duality is used in the translation invariant case, while the weighted case is partly based on the methods developed by M. Riesz in his paper of 1928 for the conjugate function operator.The third and final part derives simple and explicit formulas for reordering elements in an algebra with three generators and Lie type relations. Centralizers and centers are computed as an example of an application of the formulas.
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| 5. |
- Musonda, John, 1981-, et al.
(author)
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Reordering in a multi-parametric family of algebras
- 2019
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In: Journal of Physics. - : Institute of Physics Publishing.
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Conference paper (peer-reviewed)abstract
- This article is devoted to a multi-parametric family of associative complex algebras defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems. General reordering and nested commutator formulas for arbitrary elements in these families are presented, generalizing some well-known results in mathematics and physics. A generalization of this family in three generators is also considered.
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| 6. |
- Musonda, John, 1981-, et al.
(author)
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Reordering in noncommutative algebras associated with iterated function systems
- 2020
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In: Springer Proceedings in Mathematics and Statistics. - Cham : Springer. - 9783030418496 ; , s. 509-552
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Conference paper (peer-reviewed)abstract
- A general class of multi-parametric families of unital associative complex algebras, defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems, is considered. A generalization of these commutation relations in three generators is also considered, modifying Lie algebra type commutation relations, typical for usual differential or difference operators, to relations satisfied by more general twisted difference operators associated with general twisting maps. General reordering and nested commutator formulas for arbitrary elements in these algebras are presented, and some special cases are considered, generalizing some well-known results in mathematics and physics. © Springer Nature Switzerland AG 2020.
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| 7. |
- Musonda, John, 1981-
(author)
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Reordering in Noncommutative Algebras, Orthogonal Polynomials and Operators
- 2018
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Doctoral thesis (other academic/artistic)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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| 8. |
- Musonda, John, et al.
(author)
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Three systems of orthogonal polynomials and L2-boundedness of associated operators
- 2018
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In: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 459:1, s. 464-475
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Journal article (peer-reviewed)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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| 9. |
- Musonda, John, et al.
(author)
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Twisted difference operator representations of deformed lie type commutation relations
- 2020
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In: Springer Proceedings in Mathematics and Statistics. - Cham : Springer. - 9783030418496 ; , s. 553-573
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Conference paper (peer-reviewed)abstract
- Operator representations of deformed Lie type commutation relations, associated with group or semigroup actions of dynamical systems and iterated function systems are considered. In particular, it is shown that some multi-parameter deformed symmetric difference and multiplication operators satisfy these commutation relations. The operator representations are considered also in the context of twisted derivations. © Springer Nature Switzerland AG 2020.
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| 10. |
- Zulu, Joseph M., et al.
(author)
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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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In: Reproductive Health. - : BioMed Central. - 1742-4755. ; 15
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Journal article (peer-reviewed)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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